Research ArticleEffect of Variable Properties and Moving HeatSource on Magnetothermoelastic Problem underFractional Order Thermoelasticity

Chunbao Xiong and Ying Guo

School of Civil Engineering Tianjin University Tianjin 300072 China

Correspondence should be addressed to Ying Guo gytha yingtjueducn

Received 28 January 2016 Accepted 3 May 2016

Academic Editor Peter Majewski

Copyright copy 2016 C Xiong and Y GuoThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A one-dimensional generalized magnetothermoelastic problem of a thermoelastic rod with finite length is investigated in thecontext of the fractional order thermoelasticity The rod with variable properties which are temperature-dependent is fixed atboth ends and placed in an initial magnetic field and the rod is subjected to a moving heat source along the axial directionThe governing equations of the problem in the fractional order thermoelasticity are formulated and solved by means of Laplacetransform in tandem with its numerical inversion The distributions of the nondimensional temperature displacement and stressin the rod are obtained and illustrated graphically The effects of the temperature-dependent properties the velocity of the movingheat source the fractional order parameter and so forth on the considered variables are concerned and discussed in detail and theresults show that they significantly influence the variations of the considered variables

1 Introduction

The classical coupled thermoelasticity proposed by Biot [1]predicts an infinite speed for heat propagation which isphysically impossible To eliminate such an inherent paradoxgeneralized thermoelastic theories such as Lord and Shul-manrsquos theory (L-S) [2] and Green and Lindsayrsquos theory (G-L) [3] were developed in response The L-S theory was thefirst description of generalized thermoelasticity in which awave-type heat conduction law was postulated to replace theclassical Fourierrsquos law this wave-type law is the same as thatsuggested by Catteneo [4] and Vernotte [5] On the basis ofthese generalized models many previous researchers haveattempted to accurately capture thermomechanical behavior[6ndash11]

The generalized electromagnetothermoelastic problemin thermoelastic solids has attracted considerable researchattention due to its extensive potential applications in diversefields Examples include understanding the effects of Earthrsquosmagnetic field on seismic waves the damping of acoustic

waves in magnetic fields and the emissions of electro-magnetic radiation from nuclear devices He and Cao [12]investigated the magnetothermoelastic problem of a thinslim strip placed in a magnetic field and subjected to amoving plane of heat source Sherief and Khader [13] studiedwave propagation for a problem of an infinitely long solidconducting circular cylinder with a traction-free lateralsurface subjected to known surrounding temperatures in thepresence of a uniform magnetic field in the axis directionAbbas and Zenkour [14] presented the electromagnetother-moelastic analysis problem of an infinite functionally gradedmaterial hollow cylinder Sarkar [15] investigated an elec-tromagnetothermoelastic coupled problem for a half-spacesubjected to a thermal shock under different generalizedthermoelastic theories Pal et al [16] dealt with the problemof magnetothermoelastic interactions in a rotating mediumdue to a periodically varying heat source in the contextof generalized thermoelasticity Singh and Chakraborty [17]studied the effects of the magnetic field and initial stress onthe reflection of a planemagnetothermoelastic wave from the

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2016 Article ID 5341569 12 pageshttpdxdoiorg10115520165341569

2 Advances in Materials Science and Engineering

boundary of a solid half-space Said [18] solved a generalizedmagnetothermoelastic problem for a half-space with G-Ntheory

Fractional calculus has been used successfully to modifymany existing models of physical processes especially inthe field of heat conduction diffusion viscoelasticity solidsmechanics control theory and electricity [19ndash22] Povstenko[23] proposed a quasistatic uncoupled theory of thermoe-lasticity which is based on the heat conduction equationwith a time-fractional order derivative In 2010 Youssef [24]introduced the Riemann-Liouville fractional integral oper-ator into the generalized heat conduction and constructedthe theory of fractional order generalized thermoelasticityBy employing this theory Sarkar and Lahiri [25] concerneda two-dimensional generalized thermoelastic problem fora rotating elastic medium Youssef [26] dealt with a two-temperature generalized thermoelastic medium subjected toa moving heat source in the context of fractional ordergeneralized thermoelasticity Yu et al [27] formulated thefractional order generalized electromagnetothermoelastictheory and presented the effect of fractional order parameterYoussef and Abbas [28] solved a one-dimensional problemof an elastic half-space in the context of fractional ordergeneralized thermoelasticity Song et al [29] studied thevibration of microcantilevers during a photothermal processby coupling the theories of fractional order heat conductionand elastic waves Abbas and Youssef [30] investigated a two-dimensional problem of a porous half-space with a traction-free surface and a constant heat flux with fractional ordergeneralized thermoelasticity theory Recently a completelynew fractional order generalized thermoelasticity theory wasintroduced by Sherief et al [31] Based on this theory KothariandMukhopadhyay [32] solved an elastic half-space problemvia Laplace transform and state-space method Sherief andAbd El-Latief [33] investigated a half-space problem withvarying extent of thermal conductivity He and Guo [34]investigated a one-dimensional problem for a rod subjectedto a moving heat source Sherief and Abd El-Latief [35]investigated a one-dimensional thermal shock problem fora half-space in the context of the fractional order theory ofthermoelasticity and later solved a two-dimensional problemfor a traction-free half-space surface subjected to a heatingin the context of the same theory [36] Abbas [37] solvedthe problem of fractional order thermoelastic interaction fora material placed in a magnetic field and subjected to amoving plane of heat source Sherief and Abd El-Latief [38]solved a one-dimensional problem with a spherical cavitysubjected to a thermal shock with fractional order theory ofthermoelasticity Ma and He [39] dealt with a generalizedpiezoelectric-thermoelastic problem subjected to a movingheat source in the context of the fractional order theory ofthermoelasticity

To explore the effects of temperature-dependent prop-erties on predicting the dynamic behavior of problemsunder generalized thermoelastic theories Ezzat et al [40]investigated a problem in which modulus of elasticity wasdependent on temperature Allam et al [41] studied the elec-tromagnetothermoelastic interactions in an infinite perfectlyconducting body with a spherical cavity with G-N theory

Xiong and Tian [42] investigated the magnetothermoelasticproblem of a semi-infinite body with voids and temperature-dependent material properties placed in a transverse mag-netic field Abouelregal [43] solved a one-dimensionalboundary value problem of a semi-infinite piezoelectricmedium with temperature-dependent properties under thetheory of fractional order The problem of the generalizedthermoelastic medium for three different theories under theeffects of a gravitational field was investigated by Othmanet al [44] Pal et al [45] dealt with a thermoelastic problemof a cylindrical cavity subjected to time-dependent thermaland mechanical shocks in the context of fractional ordergeneralized thermoelasticity with the L-S model and G-Nmodel The generalized magnetothermoelastic problem ofan infinite homogeneous isotropic microstretch half-spacewith temperature-dependent material properties placed ina transverse magnetic field was investigated in the contextof different generalized thermoelastic theories by Xiong andTian [46] Wang et al [47] focused on a thermoelastic prob-lem for an elastic medium with variable properties whichthey constructed in the context of the fractional order heatconduction

To date there are relatively few works devoted to the in-vestigation of electromagnetothermoelastic problems involv-ing heat source and temperature-dependent properties in thecontext of the fractional order theory of thermoelasticityThepresent paper is devoted to our investigation of a generalizedmagnetothermoelasticmediumwith temperature-dependentproperties subjected to a moving heat source in the fractionalorder theory as proposed by Sherief et al [31]

2 Basic Equations

The generalized magnetothermoelastic governing equationsof an isotropic homogeneous conducting elastic mediumunder fractional order theory take the following forms

120590119894119895119895+ (119869 times 119861)119894 = 120588119894 (1)

119861 = 1205830119867 (2)

119869 = 1205900 (119864 + times 119861) (3)

120590119894119895= 2120583119890

119894119895+ (120582119890119896119896minus 120574120579) 120575

119894119895 (4)

119890119894119895=

1

2

(119906119894119895+ 119906119895119894) (5)

119902119894119894= minus120588119879

0 + 119876 (6)

120588120578 = 120574119890119896119896+

120588119862119864

1198790

120579 (7)

120581119894119895119879119894119895= (1 +

120591120572

0

120572

120597120572

120597119905120572) (120588119862

119864119879 + 1198790120582119894119895119890119894119895minus 119876)

0 lt 120572 le 1

(8)

Advances in Materials Science and Engineering 3

The fractional derivative is defined according to Sherief et al[31]

120597120572

120597119905120572119891 (119909 119905) =

119891 (119909 119905) minus 119891 (119909 0) 120572 997888rarr 0

1198681minus120572

120597119891 (119909 119905)

120597119905

0 lt 120572 lt 1

120597119891 (119909 119905)

120597119905

120572 = 1

(9)

where the Riemann-Liouville fractional integral 119868120572 is intro-duced as a natural generalization of the well-known 120572-foldrepeated integral 119868120572119891(119905) which is written in a convolution-type form as follows

119868120572119891 (119905) = int

119905

0

1

Γ (120572)

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 (10)

where Γ(120572) is the well-known Gamma function and 119891(119905) isLebesguersquos integrable function Equation (8) describes theprocess of heat conduction in the whole spectrum Thedifferent values of the fractional parameter within the range0 lt 120572 le 1 cover two cases of conductivity (0 lt 120572 lt 1) forweak conductivity and 120572 = 1 for normal conductivity

Here we investigate the problem of a generalized magne-tothermoelastic rod with temperature-dependent propertiessubjected to a moving heat source in the context of thefractional order theory of thermoelasticity A magnetic fieldwith constant intensity 119867 = (0119867

0 0) acts perpendicularly

to the axial direction of the rod which is fixed at both endsand subjected to a moving heat source propagating along the119909 direction The dimension along the 119909-axis is assumed to bemuch greater than those along the other two directions (119910 119911)orthogonal to the 119909-axis thus the problem can be treated asa one-dimensional problem

For the one-dimensional problem the components of theelectromagnetic induction vector are given by

119861119909= 119861119911= 0

119861119910= 12058301198670

(11)

while the components of Lorentz force119865 = 119869times119861 in themotionequation (1) can be given by

119865119909= minus12059001205832

01198672

0

119865119910= 119865119911= 0

(12)

For the one-dimensional problem the components of the dis-placement are

119906119909= 119906 (119909 119905)

119906119910= 119906119911= 0

(13)

The governing equations can be simplified as follows

120590 = (120582 + 2120583)

120597119906

120597119909

minus 120574 (119879 minus 1198790)

(120582 + 2120583)

1205972119906

1205971199092minus 120574

120597119879

120597119909

minus 12059001205832

01198672

0

120597119906

120597119905

= 120588

1205972119906

1205971199052

120581119894119895

1205972119879

1205971199092= (1 +

120591120572

0

120572

120597120572

120597119905120572)(120588119888

119864

120597119879

120597119905

+ 1205741198790

1205972119906

120597119909120597119905

minus 119876)

(14)

We consider a thermoelastic body of material with tem-perature-dependent properties in the following form [41]

120582 = 1205820119891 (119879)

120583 = 1205830119891 (119879)

120581 = 1205810119891 (119879)

120574 = 1205740119891 (119879)

(15)

where 120581 is the thermal conductivity 1205820 1205830 1205740 and 120581

0are

considered to be constants and 119891(119879) is given in a nondimen-sional function of temperature In the case of temperature-independent modulus of elasticity 119891(119879) = 1 We will assumethe following [41]

119891 (119879) = 1 minus 120577119879 (16)

where 120577 is the empirical material constantIn generalized thermoelasticity as well as in the coupled

theory only the infinitesimal temperature deviations fromthe reference temperature are considered For linearity of thegoverning partial differential equations of the problem wehave to take into account the condition |119879minus119879

0|119879071 which

gives the approximating function of 119891(119879) in the followingform

119891 (119879) asymp 1 minus 1205771198790 (17)

For convenience the following nondimensional quantitiesare also introduced

119909lowast= 11988801205780119909

119906lowast= 11988801205780119906

119905lowast= 1198882

01205780119905

1205910

lowast= 1198882

012057801205910

120579lowast=

119879 minus 1198790

1198790

120590lowast=

120590

120583

119876lowast=

119876

11989611987901198882

01205782

0

1198882

0=

120582 + 2120583

120588

1205780=

120588119862119864

119896

(18)

4 Advances in Materials Science and Engineering

In terms of the nondimensional variables in (18) (14) takethe following forms (dropping the asterisks for convenience)

120599120590 = 1205732 120597119906

120597119909

minus 119887120579

1205972119906

1205971199092minus

119887

1205732

120597120579

120597119909

= 120599(

1205972119906

1205971199052+ 120576

120597119906

120597119905

)

1205972120579

1205971199092= (1 +

120591120572

0

120572

120597120572

120597119905120572)(120599

120597120579

120597119905

+ 119892

1205972119906

120597119909120597119905

minus 119876)

(19)

where

120599 =

1

1 minus 1205771198790

1205732=

120582 + 2120583

120583

119887 =

1205741198790

120583

119892 =

120574

120599120588119862119864

120576 =

12059001205832

01198672

0

1205780(120582 + 2120583)

(20)

The rod is assumed to have reference temperature 1198790and

homogeneous initial conditions

119906 (119909 0) = (119909 0) = 0

120579 (119909 0) =120579 (119909 0) = 0

(21)

The rod is fixed at both ends with a nondimensional length 119897so the boundary conditions are

119906 (0 119905) = 119906 (119897 119905) = 0

120597120579 (0 119905)

120597119909

=

120597120579 (119897 119905)

120597119909

= 0

(22)

The heat source is moving along the 119909-axis with a constantvelocity 120592 which can be described as follows

119876 = 1198760120575 (119909 minus 120592119905) (23)

where 1198760is a constant and 120575 is the delta function

Applying the Laplace transform defined by

119871 [119891 (119905)] = 119891 (119901) = int

infin

0

119890minus119901119905119891 (119905) 119889119905 Re (119901) gt 0 (24)

to (19) with (21) we obtain

120599120590 = 1205732 119889119906

119889119909

minus 119887120579 (25)

1198892119906

1198891199092minus

119887

1205732

119889120579

119889119909

= 120599119901 (119901 + 120576) 119906 (26)

1198892120579

1198891199092= (1 +

120591120572

0

120572

119901120572)(119901120599120579 + 119892119901

119889119906

119889119909

minus 120596120599119890minus(119901120592)119909

) (27)

where

120596 =

1198760

120592

(28)

By applying the Laplace transform the boundary conditionsin (21) can be expressed as follows

119906 (0 119901) = 119906 (119897 119901) = 0

119889120579 (0 119901)

119889119909

=

119889120579 (119897 119901)

119889119909

= 0

(29)

3 Solutions in the Laplace Domain

Eliminating 120579 between (26) and (27) we obtain the followingequation satisfied by 119906

1198894119906

1198891199094minus 1198981

1198892119906

1198891199092+ 1198982119906 = 119898

3119890minus(119901120592)119909

(30)

where

1198981= (120599 +

119892119887

1205732)(1 +

120591120572

0

120572

119901120572)119901 + 120599119901 (119901 + 120576)

1198982= (1 +

120591120572

0

120572

119901120572)12059921199012(119901 + 120576)

1198983=

119887120596119901120599 (1 + (120591120572

0120572) 119901

120572)

1205732120592

(31)

The general solution of (30) is

119906 = 1198621119890minus1198961119909+ 11986221198901198961119909+ 1198623119890minus1198962119909+ 11986241198901198962119909

+ 1198625119890minus(119901120592)119909

(32)

where 119862119894(119894 = 1 2 3 4) are parameters depending on 119901 to be

determined from the boundary conditions and

1198625=

1198983

[(119901120592)4minus 1198981(119901120592)

2+ 1198982]

(33)

where 1198961and 119896

2are the roots of the following characteristic

equation

1198964minus 11989811198962+ 1198982= 0 (34)

and 1198961 1198962are given by

1198961=radic1198981+ radic119898

12minus 41198982

2

1198962=radic1198981minus radic119898

12minus 41198982

2

(35)

Similarly eliminating 119906 between (26) and (27) we obtain

1198894120579

1198891199094minus 1198981

1198892120579

1198891199092+ 1198982120579 = 119898

4119890minus(119901120592)119909

(36)

Advances in Materials Science and Engineering 5

where

1198984= 120596120599(1 +

120591120572

0

120572

119901120572)[120599119901 (119901 + 120576) minus

1199012

1205922] (37)

The general solution of (36) is

120579 = 11986211119890minus1198961119909+ 119862221198901198961119909+ 11986233119890minus1198962119909+ 119862441198901198962119909

+ 11986255119890minus(119901120592)119909

(38)

where the parameters of 119862119894119894(119894 = 1 2 3 4) are dependent on

119901Substituting 119906 from (32) and 120579 from (38) into (26) the

following relationships become clear

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198621

11986222=

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198622

11986233= minus

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198623

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198624

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(39)

To obtain the parameters of 119862119894(119894 = 1 2 3 4) and 119862

119894119894(119894 =

1 2 3 4) (32) and (38) are substituted into the equation ofboundary conditions as follows

1198621+ 1198622+ 1198623+ 1198624= minus1198625

1198621119890minus1198961119897+ 11986221198901198961119897+ 1198623119890minus1198962119897+ 11986241198901198962119897= minus1198625119890minus(119901120592)119897

minus 119862111198961+ 119862221198961minus 119862331198962+ 119862441198962= (

119901

120592

)11986255

minus 119862111198961119890minus1198961119897+ 1198622211989611198901198961119897minus 119862331198962119890minus1198962119897+ 1198624411989621198901198962119897

= (

119901

120592

)11986255119890minus(119901120592)119897

(40)

Solving (40) we obtain 119862119894(119894 = 1 2 3 4) as follows

1198621=

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198622= minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198623= minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

1198624=

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

(41)

Substituting (41) into (32) we obtain

119906 =

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625119890minus1198961119909

minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

11986251198901198961119909

minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625119890minus1198962119909

+

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

11986251198901198962119909

+ 1198625119890minus(119901120592)119909

(42)

The relationship between 119862119894and 119862

119894119894in (39) gives

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986222

= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986233=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(43)

Substituting (43) into (38) we obtain

120579 = minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

6 Advances in Materials Science and Engineering

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909+

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625119890minus(119901120592)119909

(44)

Substituting (42) and (44) into (25) yields

120590 = minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909minus 1205732120592 (119901 + 120576) 119862

5119890minus(119901120592)119909

(45)

4 Numerical Inversion of the Transforms

In order to determine the nondimensional temperaturenondimensional displacement and nondimensional stressin the rod we need to invert the parameters of 120579 119906 and120590 from the Laplace domain Unfortunately the obtainedsolutions in the Laplace domain are too complicated tobe inverted analytically thus a feasible numerical methodthat is the Riemann-sum approximation method is used tocomplete the inversion Accordingly any function 119891(119909 119901) inthe Laplace domain can be inverted to the time domain asfollows [48]

119891 (119909 119905)

=

119890120573119905

119905

[

1

2

119891 (119909 120573) + Re119873

sum

119899=1

119891(119909 120573 +

119894119899120587

119905

) (minus1)119899]

(46)

where Re is the real part and 119894 is the imaginary number unitFor faster convergence numerous numerical experimentshave shown that the value of 120573 satisfies the relation 120573119905 asymp 47

[48]

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 1 Nondimensional temperature distribution for different 119905values

5 Numerical Results and Discussions

In terms of the Riemann-sum approximation defined in (46)numerical Laplace inversion is implemented to obtain thenondimensional temperature displacement and stress in therod in the time domain For the purposes of simulationthe thermoelastic material is specified as copper and theparameters are

120582 = 776 times 1010Nmminus2

120583 = 386 times 1010Nmminus2

120588 = 8954 kgmminus3

120572119905= 178 times 10

minus5 Kminus1

119862119864= 3831 JKgminus1 Kminus1

120581 = 386Wmminus1 Kminus1

1198790= 293K

120577 = 00005Kminus1

(47)

The other constants are

1198760= 10

1205910= 005

119897 = 10

(48)

Numerical calculation is carried out for the following fivecases

In Case 1 we investigate the nondimensional tempera-ture displacement and stress varying with time as shown inFigures 1ndash3 with the moving heat source velocity fractionalorder parameter value of temperature-dependent properties

Advances in Materials Science and Engineering 7

1 2 3 4 5 6 7 8 9 10minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 2 Nondimensional stress distribution for different 119905 values

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 3 Nondimensional displacement distribution for different 119905values

and magnetic field set as constants (ie 120592 = 20 120572 = 025120599 = 05 120576 = 5) In Case 2 the considered variables varyingwith different moving heat source velocity are investigatedwhen 119905 = 15 120572 = 025 120599 = 05 and 120576 = 5 (Figures 4ndash6) In Case 3 we investigate how the considered variablesvary with different magnetic fields when 119905 = 15 120592 = 20120599 = 05 and 120572 = 025 (Figures 7ndash9) Case 4 involvesthe considered variables varying with different temperature-dependent properties when 119905 = 15 120592 = 20 120572 = 025 and120576 = 5 (Figures 10ndash12) and in Case 5 we investigate howthe considered variables vary with different fractional orderparameters when 119905 = 15 120592 = 20 120599 = 05 and 120576 = 5 (Figures13ndash15)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 4 Nondimensional temperature distribution for different 120592values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00N

ondi

men

siona

l stre

ss 120590

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 5 Nondimensional stress distribution for different 120592 values

In Figures 1ndash3 (Case 1) the solid line dash line and dotline refer to 119905 = 15 119905 = 20 and 119905 = 25 respectively Figure 1shows that the nondimensional temperature increases as thetime increases At location 119909 = 120592119905 the heat source releases itsmaximum energy which leads to a peak value As shown inFigure 2 the nondimensional stress in the rod is compressiveDue to the fixed ends thermal expansion deformation isrestrained in both ends and leads to the occurrence ofcompressive thermal stress in the rod The absolute valueof stress increases with the passage of time As shown inFigure 3 the nondimensional displacement also increasesas time progresses Due to the applied moving heat sourcethe rod undergoes thermal expansion deformation With thepassage of time the heat disturbed region enlarges so thatthermal expansion deformation evolves along the rod

8 Advances in Materials Science and Engineering

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 6 Nondimensional displacement distribution for different 120592values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 7 Nondimensional temperature distribution for different 120576values

In Figures 4ndash6 (Case 2) the solid line dash line anddot line refer to 120592 = 20 120592 = 30 and 120592 = 40 respec-tively As shown in Figure 4 the nondimensional temper-ature decreases with the increasing of the moving heat re-source velocity In a given period the energy that the heatsource can release is constant however the intensity ofthe released energy per unit length decreases as the sourcespeed increases It is also clear in Figures 5 and 6 that themagnitudes of the nondimensional stress and displacementdecrease as the moving heat source velocity increases whichis the result of reduction in heat energy intensity per unitlength at greater velocities

In Figures 7ndash9 (Case 3) the solid line dash line and dotline refer to 120576 = 0 120576 = 50 and 120576 = 100 respectively

1 2 3 4 5 6 7 8 9 10minus07

minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 8 Nondimensional stress distribution for different 120576 values

0 1 2 3 4 5 6 7 8 9 100000

0005

0010

0015

0020

0025

0030N

ondi

men

siona

l disp

lace

men

t u

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 9 Nondimensional displacement distribution for different 120576values

In Figure 7 the three lines overlap because the magneticfield has no effect on the temperature As shown in Figure 8the absolute value of stress increases as the 120576 value increasesprior to the point where the curves intersect then afterthe intersection the absolute value of stress decreases as theapplied magnetic field value increases Figure 9 for Case 3shows that nondimensional displacement decreases as themagnetic field increases The largest displacement value is inthe case of 120576 = 0 namely in the absence of the magnetic fieldThis indicates that themagnetic field acts to damp the thermalexpansion deformation of the rod

In Figures 10ndash12 (Case 4) the solid line dash line anddot line refer to 120599 = 05 120599 = 075 and 120599 = 10 respectivelyAs shown in Figure 10 the temperature increases as the 120599

Advances in Materials Science and Engineering 9

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 10 Nondimensional temperature distribution for different 120599values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 11 Nondimensional stress distribution for different 120599 values

value increases before the three curves intersect at whichpoint it decreases as the temperature-dependent propertiesincrease Figures 11 and 12 show where the displacementand absolute value of stress decrease as the temperature-dependent properties increase

Figures 13ndash15 (Case 5) show the variations of the nondi-mensional temperature stress and displacement respec-tively which demonstrate the effects of the fractional orderparameter on the variations of the considered variables Tothis end a series of values of the fractional order parameter120572 within the region (0 1] are tested through numericalcalculation As representation of the effect of 120572 three typicalvalues of 120572 120572 = 025 120572 = 05 and 120572 = 10 are consideredFigures 13ndash15 show that temperature displacement and

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 12 Nondimensional displacement distribution for different120599 values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 13 Nondimensional temperature distribution for different 120572values

absolute stress value all decrease as the fractional order valueincreases prior to the intersection of the three curves Afterthe intersection however all considered variables increasewith the increasing of the fractional order parameter

6 Conclusions

A generalized magnetothermoelastic rod with temperature-dependent properties subjected to a moving heat sourceis investigated in the fractional order theory The resultsprovided the following most notable conclusions

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

2 Advances in Materials Science and Engineering

boundary of a solid half-space Said [18] solved a generalizedmagnetothermoelastic problem for a half-space with G-Ntheory

Fractional calculus has been used successfully to modifymany existing models of physical processes especially inthe field of heat conduction diffusion viscoelasticity solidsmechanics control theory and electricity [19ndash22] Povstenko[23] proposed a quasistatic uncoupled theory of thermoe-lasticity which is based on the heat conduction equationwith a time-fractional order derivative In 2010 Youssef [24]introduced the Riemann-Liouville fractional integral oper-ator into the generalized heat conduction and constructedthe theory of fractional order generalized thermoelasticityBy employing this theory Sarkar and Lahiri [25] concerneda two-dimensional generalized thermoelastic problem fora rotating elastic medium Youssef [26] dealt with a two-temperature generalized thermoelastic medium subjected toa moving heat source in the context of fractional ordergeneralized thermoelasticity Yu et al [27] formulated thefractional order generalized electromagnetothermoelastictheory and presented the effect of fractional order parameterYoussef and Abbas [28] solved a one-dimensional problemof an elastic half-space in the context of fractional ordergeneralized thermoelasticity Song et al [29] studied thevibration of microcantilevers during a photothermal processby coupling the theories of fractional order heat conductionand elastic waves Abbas and Youssef [30] investigated a two-dimensional problem of a porous half-space with a traction-free surface and a constant heat flux with fractional ordergeneralized thermoelasticity theory Recently a completelynew fractional order generalized thermoelasticity theory wasintroduced by Sherief et al [31] Based on this theory KothariandMukhopadhyay [32] solved an elastic half-space problemvia Laplace transform and state-space method Sherief andAbd El-Latief [33] investigated a half-space problem withvarying extent of thermal conductivity He and Guo [34]investigated a one-dimensional problem for a rod subjectedto a moving heat source Sherief and Abd El-Latief [35]investigated a one-dimensional thermal shock problem fora half-space in the context of the fractional order theory ofthermoelasticity and later solved a two-dimensional problemfor a traction-free half-space surface subjected to a heatingin the context of the same theory [36] Abbas [37] solvedthe problem of fractional order thermoelastic interaction fora material placed in a magnetic field and subjected to amoving plane of heat source Sherief and Abd El-Latief [38]solved a one-dimensional problem with a spherical cavitysubjected to a thermal shock with fractional order theory ofthermoelasticity Ma and He [39] dealt with a generalizedpiezoelectric-thermoelastic problem subjected to a movingheat source in the context of the fractional order theory ofthermoelasticity

To explore the effects of temperature-dependent prop-erties on predicting the dynamic behavior of problemsunder generalized thermoelastic theories Ezzat et al [40]investigated a problem in which modulus of elasticity wasdependent on temperature Allam et al [41] studied the elec-tromagnetothermoelastic interactions in an infinite perfectlyconducting body with a spherical cavity with G-N theory

Xiong and Tian [42] investigated the magnetothermoelasticproblem of a semi-infinite body with voids and temperature-dependent material properties placed in a transverse mag-netic field Abouelregal [43] solved a one-dimensionalboundary value problem of a semi-infinite piezoelectricmedium with temperature-dependent properties under thetheory of fractional order The problem of the generalizedthermoelastic medium for three different theories under theeffects of a gravitational field was investigated by Othmanet al [44] Pal et al [45] dealt with a thermoelastic problemof a cylindrical cavity subjected to time-dependent thermaland mechanical shocks in the context of fractional ordergeneralized thermoelasticity with the L-S model and G-Nmodel The generalized magnetothermoelastic problem ofan infinite homogeneous isotropic microstretch half-spacewith temperature-dependent material properties placed ina transverse magnetic field was investigated in the contextof different generalized thermoelastic theories by Xiong andTian [46] Wang et al [47] focused on a thermoelastic prob-lem for an elastic medium with variable properties whichthey constructed in the context of the fractional order heatconduction

To date there are relatively few works devoted to the in-vestigation of electromagnetothermoelastic problems involv-ing heat source and temperature-dependent properties in thecontext of the fractional order theory of thermoelasticityThepresent paper is devoted to our investigation of a generalizedmagnetothermoelasticmediumwith temperature-dependentproperties subjected to a moving heat source in the fractionalorder theory as proposed by Sherief et al [31]

2 Basic Equations

The generalized magnetothermoelastic governing equationsof an isotropic homogeneous conducting elastic mediumunder fractional order theory take the following forms

120590119894119895119895+ (119869 times 119861)119894 = 120588119894 (1)

119861 = 1205830119867 (2)

119869 = 1205900 (119864 + times 119861) (3)

120590119894119895= 2120583119890

119894119895+ (120582119890119896119896minus 120574120579) 120575

119894119895 (4)

119890119894119895=

1

2

(119906119894119895+ 119906119895119894) (5)

119902119894119894= minus120588119879

0 + 119876 (6)

120588120578 = 120574119890119896119896+

120588119862119864

1198790

120579 (7)

120581119894119895119879119894119895= (1 +

120591120572

0

120572

120597120572

120597119905120572) (120588119862

119864119879 + 1198790120582119894119895119890119894119895minus 119876)

0 lt 120572 le 1

(8)

Advances in Materials Science and Engineering 3

The fractional derivative is defined according to Sherief et al[31]

120597120572

120597119905120572119891 (119909 119905) =

119891 (119909 119905) minus 119891 (119909 0) 120572 997888rarr 0

1198681minus120572

120597119891 (119909 119905)

120597119905

0 lt 120572 lt 1

120597119891 (119909 119905)

120597119905

120572 = 1

(9)

where the Riemann-Liouville fractional integral 119868120572 is intro-duced as a natural generalization of the well-known 120572-foldrepeated integral 119868120572119891(119905) which is written in a convolution-type form as follows

119868120572119891 (119905) = int

119905

0

1

Γ (120572)

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 (10)

where Γ(120572) is the well-known Gamma function and 119891(119905) isLebesguersquos integrable function Equation (8) describes theprocess of heat conduction in the whole spectrum Thedifferent values of the fractional parameter within the range0 lt 120572 le 1 cover two cases of conductivity (0 lt 120572 lt 1) forweak conductivity and 120572 = 1 for normal conductivity

Here we investigate the problem of a generalized magne-tothermoelastic rod with temperature-dependent propertiessubjected to a moving heat source in the context of thefractional order theory of thermoelasticity A magnetic fieldwith constant intensity 119867 = (0119867

0 0) acts perpendicularly

to the axial direction of the rod which is fixed at both endsand subjected to a moving heat source propagating along the119909 direction The dimension along the 119909-axis is assumed to bemuch greater than those along the other two directions (119910 119911)orthogonal to the 119909-axis thus the problem can be treated asa one-dimensional problem

For the one-dimensional problem the components of theelectromagnetic induction vector are given by

119861119909= 119861119911= 0

119861119910= 12058301198670

(11)

while the components of Lorentz force119865 = 119869times119861 in themotionequation (1) can be given by

119865119909= minus12059001205832

01198672

0

119865119910= 119865119911= 0

(12)

For the one-dimensional problem the components of the dis-placement are

119906119909= 119906 (119909 119905)

119906119910= 119906119911= 0

(13)

The governing equations can be simplified as follows

120590 = (120582 + 2120583)

120597119906

120597119909

minus 120574 (119879 minus 1198790)

(120582 + 2120583)

1205972119906

1205971199092minus 120574

120597119879

120597119909

minus 12059001205832

01198672

0

120597119906

120597119905

= 120588

1205972119906

1205971199052

120581119894119895

1205972119879

1205971199092= (1 +

120591120572

0

120572

120597120572

120597119905120572)(120588119888

119864

120597119879

120597119905

+ 1205741198790

1205972119906

120597119909120597119905

minus 119876)

(14)

We consider a thermoelastic body of material with tem-perature-dependent properties in the following form [41]

120582 = 1205820119891 (119879)

120583 = 1205830119891 (119879)

120581 = 1205810119891 (119879)

120574 = 1205740119891 (119879)

(15)

where 120581 is the thermal conductivity 1205820 1205830 1205740 and 120581

0are

considered to be constants and 119891(119879) is given in a nondimen-sional function of temperature In the case of temperature-independent modulus of elasticity 119891(119879) = 1 We will assumethe following [41]

119891 (119879) = 1 minus 120577119879 (16)

where 120577 is the empirical material constantIn generalized thermoelasticity as well as in the coupled

theory only the infinitesimal temperature deviations fromthe reference temperature are considered For linearity of thegoverning partial differential equations of the problem wehave to take into account the condition |119879minus119879

0|119879071 which

gives the approximating function of 119891(119879) in the followingform

119891 (119879) asymp 1 minus 1205771198790 (17)

For convenience the following nondimensional quantitiesare also introduced

119909lowast= 11988801205780119909

119906lowast= 11988801205780119906

119905lowast= 1198882

01205780119905

1205910

lowast= 1198882

012057801205910

120579lowast=

119879 minus 1198790

1198790

120590lowast=

120590

120583

119876lowast=

119876

11989611987901198882

01205782

0

1198882

0=

120582 + 2120583

120588

1205780=

120588119862119864

119896

(18)

4 Advances in Materials Science and Engineering

In terms of the nondimensional variables in (18) (14) takethe following forms (dropping the asterisks for convenience)

120599120590 = 1205732 120597119906

120597119909

minus 119887120579

1205972119906

1205971199092minus

119887

1205732

120597120579

120597119909

= 120599(

1205972119906

1205971199052+ 120576

120597119906

120597119905

)

1205972120579

1205971199092= (1 +

120591120572

0

120572

120597120572

120597119905120572)(120599

120597120579

120597119905

+ 119892

1205972119906

120597119909120597119905

minus 119876)

(19)

where

120599 =

1

1 minus 1205771198790

1205732=

120582 + 2120583

120583

119887 =

1205741198790

120583

119892 =

120574

120599120588119862119864

120576 =

12059001205832

01198672

0

1205780(120582 + 2120583)

(20)

The rod is assumed to have reference temperature 1198790and

homogeneous initial conditions

119906 (119909 0) = (119909 0) = 0

120579 (119909 0) =120579 (119909 0) = 0

(21)

The rod is fixed at both ends with a nondimensional length 119897so the boundary conditions are

119906 (0 119905) = 119906 (119897 119905) = 0

120597120579 (0 119905)

120597119909

=

120597120579 (119897 119905)

120597119909

= 0

(22)

The heat source is moving along the 119909-axis with a constantvelocity 120592 which can be described as follows

119876 = 1198760120575 (119909 minus 120592119905) (23)

where 1198760is a constant and 120575 is the delta function

Applying the Laplace transform defined by

119871 [119891 (119905)] = 119891 (119901) = int

infin

0

119890minus119901119905119891 (119905) 119889119905 Re (119901) gt 0 (24)

to (19) with (21) we obtain

120599120590 = 1205732 119889119906

119889119909

minus 119887120579 (25)

1198892119906

1198891199092minus

119887

1205732

119889120579

119889119909

= 120599119901 (119901 + 120576) 119906 (26)

1198892120579

1198891199092= (1 +

120591120572

0

120572

119901120572)(119901120599120579 + 119892119901

119889119906

119889119909

minus 120596120599119890minus(119901120592)119909

) (27)

where

120596 =

1198760

120592

(28)

By applying the Laplace transform the boundary conditionsin (21) can be expressed as follows

119906 (0 119901) = 119906 (119897 119901) = 0

119889120579 (0 119901)

119889119909

=

119889120579 (119897 119901)

119889119909

= 0

(29)

3 Solutions in the Laplace Domain

Eliminating 120579 between (26) and (27) we obtain the followingequation satisfied by 119906

1198894119906

1198891199094minus 1198981

1198892119906

1198891199092+ 1198982119906 = 119898

3119890minus(119901120592)119909

(30)

where

1198981= (120599 +

119892119887

1205732)(1 +

120591120572

0

120572

119901120572)119901 + 120599119901 (119901 + 120576)

1198982= (1 +

120591120572

0

120572

119901120572)12059921199012(119901 + 120576)

1198983=

119887120596119901120599 (1 + (120591120572

0120572) 119901

120572)

1205732120592

(31)

The general solution of (30) is

119906 = 1198621119890minus1198961119909+ 11986221198901198961119909+ 1198623119890minus1198962119909+ 11986241198901198962119909

+ 1198625119890minus(119901120592)119909

(32)

where 119862119894(119894 = 1 2 3 4) are parameters depending on 119901 to be

determined from the boundary conditions and

1198625=

1198983

[(119901120592)4minus 1198981(119901120592)

2+ 1198982]

(33)

where 1198961and 119896

2are the roots of the following characteristic

equation

1198964minus 11989811198962+ 1198982= 0 (34)

and 1198961 1198962are given by

1198961=radic1198981+ radic119898

12minus 41198982

2

1198962=radic1198981minus radic119898

12minus 41198982

2

(35)

Similarly eliminating 119906 between (26) and (27) we obtain

1198894120579

1198891199094minus 1198981

1198892120579

1198891199092+ 1198982120579 = 119898

4119890minus(119901120592)119909

(36)

Advances in Materials Science and Engineering 5

where

1198984= 120596120599(1 +

120591120572

0

120572

119901120572)[120599119901 (119901 + 120576) minus

1199012

1205922] (37)

The general solution of (36) is

120579 = 11986211119890minus1198961119909+ 119862221198901198961119909+ 11986233119890minus1198962119909+ 119862441198901198962119909

+ 11986255119890minus(119901120592)119909

(38)

where the parameters of 119862119894119894(119894 = 1 2 3 4) are dependent on

119901Substituting 119906 from (32) and 120579 from (38) into (26) the

following relationships become clear

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198621

11986222=

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198622

11986233= minus

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198623

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198624

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(39)

To obtain the parameters of 119862119894(119894 = 1 2 3 4) and 119862

119894119894(119894 =

1 2 3 4) (32) and (38) are substituted into the equation ofboundary conditions as follows

1198621+ 1198622+ 1198623+ 1198624= minus1198625

1198621119890minus1198961119897+ 11986221198901198961119897+ 1198623119890minus1198962119897+ 11986241198901198962119897= minus1198625119890minus(119901120592)119897

minus 119862111198961+ 119862221198961minus 119862331198962+ 119862441198962= (

119901

120592

)11986255

minus 119862111198961119890minus1198961119897+ 1198622211989611198901198961119897minus 119862331198962119890minus1198962119897+ 1198624411989621198901198962119897

= (

119901

120592

)11986255119890minus(119901120592)119897

(40)

Solving (40) we obtain 119862119894(119894 = 1 2 3 4) as follows

1198621=

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198622= minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198623= minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

1198624=

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

(41)

Substituting (41) into (32) we obtain

119906 =

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625119890minus1198961119909

minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

11986251198901198961119909

minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625119890minus1198962119909

+

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

11986251198901198962119909

+ 1198625119890minus(119901120592)119909

(42)

The relationship between 119862119894and 119862

119894119894in (39) gives

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986222

= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986233=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(43)

Substituting (43) into (38) we obtain

120579 = minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

6 Advances in Materials Science and Engineering

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909+

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625119890minus(119901120592)119909

(44)

Substituting (42) and (44) into (25) yields

120590 = minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909minus 1205732120592 (119901 + 120576) 119862

5119890minus(119901120592)119909

(45)

4 Numerical Inversion of the Transforms

In order to determine the nondimensional temperaturenondimensional displacement and nondimensional stressin the rod we need to invert the parameters of 120579 119906 and120590 from the Laplace domain Unfortunately the obtainedsolutions in the Laplace domain are too complicated tobe inverted analytically thus a feasible numerical methodthat is the Riemann-sum approximation method is used tocomplete the inversion Accordingly any function 119891(119909 119901) inthe Laplace domain can be inverted to the time domain asfollows [48]

119891 (119909 119905)

=

119890120573119905

119905

[

1

2

119891 (119909 120573) + Re119873

sum

119899=1

119891(119909 120573 +

119894119899120587

119905

) (minus1)119899]

(46)

where Re is the real part and 119894 is the imaginary number unitFor faster convergence numerous numerical experimentshave shown that the value of 120573 satisfies the relation 120573119905 asymp 47

[48]

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 1 Nondimensional temperature distribution for different 119905values

5 Numerical Results and Discussions

In terms of the Riemann-sum approximation defined in (46)numerical Laplace inversion is implemented to obtain thenondimensional temperature displacement and stress in therod in the time domain For the purposes of simulationthe thermoelastic material is specified as copper and theparameters are

120582 = 776 times 1010Nmminus2

120583 = 386 times 1010Nmminus2

120588 = 8954 kgmminus3

120572119905= 178 times 10

minus5 Kminus1

119862119864= 3831 JKgminus1 Kminus1

120581 = 386Wmminus1 Kminus1

1198790= 293K

120577 = 00005Kminus1

(47)

The other constants are

1198760= 10

1205910= 005

119897 = 10

(48)

Numerical calculation is carried out for the following fivecases

In Case 1 we investigate the nondimensional tempera-ture displacement and stress varying with time as shown inFigures 1ndash3 with the moving heat source velocity fractionalorder parameter value of temperature-dependent properties

Advances in Materials Science and Engineering 7

1 2 3 4 5 6 7 8 9 10minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 2 Nondimensional stress distribution for different 119905 values

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 3 Nondimensional displacement distribution for different 119905values

and magnetic field set as constants (ie 120592 = 20 120572 = 025120599 = 05 120576 = 5) In Case 2 the considered variables varyingwith different moving heat source velocity are investigatedwhen 119905 = 15 120572 = 025 120599 = 05 and 120576 = 5 (Figures 4ndash6) In Case 3 we investigate how the considered variablesvary with different magnetic fields when 119905 = 15 120592 = 20120599 = 05 and 120572 = 025 (Figures 7ndash9) Case 4 involvesthe considered variables varying with different temperature-dependent properties when 119905 = 15 120592 = 20 120572 = 025 and120576 = 5 (Figures 10ndash12) and in Case 5 we investigate howthe considered variables vary with different fractional orderparameters when 119905 = 15 120592 = 20 120599 = 05 and 120576 = 5 (Figures13ndash15)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 4 Nondimensional temperature distribution for different 120592values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00N

ondi

men

siona

l stre

ss 120590

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 5 Nondimensional stress distribution for different 120592 values

In Figures 1ndash3 (Case 1) the solid line dash line and dotline refer to 119905 = 15 119905 = 20 and 119905 = 25 respectively Figure 1shows that the nondimensional temperature increases as thetime increases At location 119909 = 120592119905 the heat source releases itsmaximum energy which leads to a peak value As shown inFigure 2 the nondimensional stress in the rod is compressiveDue to the fixed ends thermal expansion deformation isrestrained in both ends and leads to the occurrence ofcompressive thermal stress in the rod The absolute valueof stress increases with the passage of time As shown inFigure 3 the nondimensional displacement also increasesas time progresses Due to the applied moving heat sourcethe rod undergoes thermal expansion deformation With thepassage of time the heat disturbed region enlarges so thatthermal expansion deformation evolves along the rod

8 Advances in Materials Science and Engineering

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 6 Nondimensional displacement distribution for different 120592values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 7 Nondimensional temperature distribution for different 120576values

In Figures 4ndash6 (Case 2) the solid line dash line anddot line refer to 120592 = 20 120592 = 30 and 120592 = 40 respec-tively As shown in Figure 4 the nondimensional temper-ature decreases with the increasing of the moving heat re-source velocity In a given period the energy that the heatsource can release is constant however the intensity ofthe released energy per unit length decreases as the sourcespeed increases It is also clear in Figures 5 and 6 that themagnitudes of the nondimensional stress and displacementdecrease as the moving heat source velocity increases whichis the result of reduction in heat energy intensity per unitlength at greater velocities

In Figures 7ndash9 (Case 3) the solid line dash line and dotline refer to 120576 = 0 120576 = 50 and 120576 = 100 respectively

1 2 3 4 5 6 7 8 9 10minus07

minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 8 Nondimensional stress distribution for different 120576 values

0 1 2 3 4 5 6 7 8 9 100000

0005

0010

0015

0020

0025

0030N

ondi

men

siona

l disp

lace

men

t u

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 9 Nondimensional displacement distribution for different 120576values

In Figure 7 the three lines overlap because the magneticfield has no effect on the temperature As shown in Figure 8the absolute value of stress increases as the 120576 value increasesprior to the point where the curves intersect then afterthe intersection the absolute value of stress decreases as theapplied magnetic field value increases Figure 9 for Case 3shows that nondimensional displacement decreases as themagnetic field increases The largest displacement value is inthe case of 120576 = 0 namely in the absence of the magnetic fieldThis indicates that themagnetic field acts to damp the thermalexpansion deformation of the rod

In Figures 10ndash12 (Case 4) the solid line dash line anddot line refer to 120599 = 05 120599 = 075 and 120599 = 10 respectivelyAs shown in Figure 10 the temperature increases as the 120599

Advances in Materials Science and Engineering 9

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 10 Nondimensional temperature distribution for different 120599values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 11 Nondimensional stress distribution for different 120599 values

value increases before the three curves intersect at whichpoint it decreases as the temperature-dependent propertiesincrease Figures 11 and 12 show where the displacementand absolute value of stress decrease as the temperature-dependent properties increase

Figures 13ndash15 (Case 5) show the variations of the nondi-mensional temperature stress and displacement respec-tively which demonstrate the effects of the fractional orderparameter on the variations of the considered variables Tothis end a series of values of the fractional order parameter120572 within the region (0 1] are tested through numericalcalculation As representation of the effect of 120572 three typicalvalues of 120572 120572 = 025 120572 = 05 and 120572 = 10 are consideredFigures 13ndash15 show that temperature displacement and

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 12 Nondimensional displacement distribution for different120599 values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 13 Nondimensional temperature distribution for different 120572values

absolute stress value all decrease as the fractional order valueincreases prior to the intersection of the three curves Afterthe intersection however all considered variables increasewith the increasing of the fractional order parameter

6 Conclusions

A generalized magnetothermoelastic rod with temperature-dependent properties subjected to a moving heat sourceis investigated in the fractional order theory The resultsprovided the following most notable conclusions

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

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NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 3

The fractional derivative is defined according to Sherief et al[31]

120597120572

120597119905120572119891 (119909 119905) =

119891 (119909 119905) minus 119891 (119909 0) 120572 997888rarr 0

1198681minus120572

120597119891 (119909 119905)

120597119905

0 lt 120572 lt 1

120597119891 (119909 119905)

120597119905

120572 = 1

(9)

where the Riemann-Liouville fractional integral 119868120572 is intro-duced as a natural generalization of the well-known 120572-foldrepeated integral 119868120572119891(119905) which is written in a convolution-type form as follows

119868120572119891 (119905) = int

119905

0

1

Γ (120572)

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 (10)

where Γ(120572) is the well-known Gamma function and 119891(119905) isLebesguersquos integrable function Equation (8) describes theprocess of heat conduction in the whole spectrum Thedifferent values of the fractional parameter within the range0 lt 120572 le 1 cover two cases of conductivity (0 lt 120572 lt 1) forweak conductivity and 120572 = 1 for normal conductivity

Here we investigate the problem of a generalized magne-tothermoelastic rod with temperature-dependent propertiessubjected to a moving heat source in the context of thefractional order theory of thermoelasticity A magnetic fieldwith constant intensity 119867 = (0119867

0 0) acts perpendicularly

to the axial direction of the rod which is fixed at both endsand subjected to a moving heat source propagating along the119909 direction The dimension along the 119909-axis is assumed to bemuch greater than those along the other two directions (119910 119911)orthogonal to the 119909-axis thus the problem can be treated asa one-dimensional problem

For the one-dimensional problem the components of theelectromagnetic induction vector are given by

119861119909= 119861119911= 0

119861119910= 12058301198670

(11)

while the components of Lorentz force119865 = 119869times119861 in themotionequation (1) can be given by

119865119909= minus12059001205832

01198672

0

119865119910= 119865119911= 0

(12)

For the one-dimensional problem the components of the dis-placement are

119906119909= 119906 (119909 119905)

119906119910= 119906119911= 0

(13)

The governing equations can be simplified as follows

120590 = (120582 + 2120583)

120597119906

120597119909

minus 120574 (119879 minus 1198790)

(120582 + 2120583)

1205972119906

1205971199092minus 120574

120597119879

120597119909

minus 12059001205832

01198672

0

120597119906

120597119905

= 120588

1205972119906

1205971199052

120581119894119895

1205972119879

1205971199092= (1 +

120591120572

0

120572

120597120572

120597119905120572)(120588119888

119864

120597119879

120597119905

+ 1205741198790

1205972119906

120597119909120597119905

minus 119876)

(14)

We consider a thermoelastic body of material with tem-perature-dependent properties in the following form [41]

120582 = 1205820119891 (119879)

120583 = 1205830119891 (119879)

120581 = 1205810119891 (119879)

120574 = 1205740119891 (119879)

(15)

where 120581 is the thermal conductivity 1205820 1205830 1205740 and 120581

0are

considered to be constants and 119891(119879) is given in a nondimen-sional function of temperature In the case of temperature-independent modulus of elasticity 119891(119879) = 1 We will assumethe following [41]

119891 (119879) = 1 minus 120577119879 (16)

where 120577 is the empirical material constantIn generalized thermoelasticity as well as in the coupled

theory only the infinitesimal temperature deviations fromthe reference temperature are considered For linearity of thegoverning partial differential equations of the problem wehave to take into account the condition |119879minus119879

0|119879071 which

gives the approximating function of 119891(119879) in the followingform

119891 (119879) asymp 1 minus 1205771198790 (17)

For convenience the following nondimensional quantitiesare also introduced

119909lowast= 11988801205780119909

119906lowast= 11988801205780119906

119905lowast= 1198882

01205780119905

1205910

lowast= 1198882

012057801205910

120579lowast=

119879 minus 1198790

1198790

120590lowast=

120590

120583

119876lowast=

119876

11989611987901198882

01205782

0

1198882

0=

120582 + 2120583

120588

1205780=

120588119862119864

119896

(18)

4 Advances in Materials Science and Engineering

In terms of the nondimensional variables in (18) (14) takethe following forms (dropping the asterisks for convenience)

120599120590 = 1205732 120597119906

120597119909

minus 119887120579

1205972119906

1205971199092minus

119887

1205732

120597120579

120597119909

= 120599(

1205972119906

1205971199052+ 120576

120597119906

120597119905

)

1205972120579

1205971199092= (1 +

120591120572

0

120572

120597120572

120597119905120572)(120599

120597120579

120597119905

+ 119892

1205972119906

120597119909120597119905

minus 119876)

(19)

where

120599 =

1

1 minus 1205771198790

1205732=

120582 + 2120583

120583

119887 =

1205741198790

120583

119892 =

120574

120599120588119862119864

120576 =

12059001205832

01198672

0

1205780(120582 + 2120583)

(20)

The rod is assumed to have reference temperature 1198790and

homogeneous initial conditions

119906 (119909 0) = (119909 0) = 0

120579 (119909 0) =120579 (119909 0) = 0

(21)

The rod is fixed at both ends with a nondimensional length 119897so the boundary conditions are

119906 (0 119905) = 119906 (119897 119905) = 0

120597120579 (0 119905)

120597119909

=

120597120579 (119897 119905)

120597119909

= 0

(22)

The heat source is moving along the 119909-axis with a constantvelocity 120592 which can be described as follows

119876 = 1198760120575 (119909 minus 120592119905) (23)

where 1198760is a constant and 120575 is the delta function

Applying the Laplace transform defined by

119871 [119891 (119905)] = 119891 (119901) = int

infin

0

119890minus119901119905119891 (119905) 119889119905 Re (119901) gt 0 (24)

to (19) with (21) we obtain

120599120590 = 1205732 119889119906

119889119909

minus 119887120579 (25)

1198892119906

1198891199092minus

119887

1205732

119889120579

119889119909

= 120599119901 (119901 + 120576) 119906 (26)

1198892120579

1198891199092= (1 +

120591120572

0

120572

119901120572)(119901120599120579 + 119892119901

119889119906

119889119909

minus 120596120599119890minus(119901120592)119909

) (27)

where

120596 =

1198760

120592

(28)

By applying the Laplace transform the boundary conditionsin (21) can be expressed as follows

119906 (0 119901) = 119906 (119897 119901) = 0

119889120579 (0 119901)

119889119909

=

119889120579 (119897 119901)

119889119909

= 0

(29)

3 Solutions in the Laplace Domain

Eliminating 120579 between (26) and (27) we obtain the followingequation satisfied by 119906

1198894119906

1198891199094minus 1198981

1198892119906

1198891199092+ 1198982119906 = 119898

3119890minus(119901120592)119909

(30)

where

1198981= (120599 +

119892119887

1205732)(1 +

120591120572

0

120572

119901120572)119901 + 120599119901 (119901 + 120576)

1198982= (1 +

120591120572

0

120572

119901120572)12059921199012(119901 + 120576)

1198983=

119887120596119901120599 (1 + (120591120572

0120572) 119901

120572)

1205732120592

(31)

The general solution of (30) is

119906 = 1198621119890minus1198961119909+ 11986221198901198961119909+ 1198623119890minus1198962119909+ 11986241198901198962119909

+ 1198625119890minus(119901120592)119909

(32)

where 119862119894(119894 = 1 2 3 4) are parameters depending on 119901 to be

determined from the boundary conditions and

1198625=

1198983

[(119901120592)4minus 1198981(119901120592)

2+ 1198982]

(33)

where 1198961and 119896

2are the roots of the following characteristic

equation

1198964minus 11989811198962+ 1198982= 0 (34)

and 1198961 1198962are given by

1198961=radic1198981+ radic119898

12minus 41198982

2

1198962=radic1198981minus radic119898

12minus 41198982

2

(35)

Similarly eliminating 119906 between (26) and (27) we obtain

1198894120579

1198891199094minus 1198981

1198892120579

1198891199092+ 1198982120579 = 119898

4119890minus(119901120592)119909

(36)

Advances in Materials Science and Engineering 5

where

1198984= 120596120599(1 +

120591120572

0

120572

119901120572)[120599119901 (119901 + 120576) minus

1199012

1205922] (37)

The general solution of (36) is

120579 = 11986211119890minus1198961119909+ 119862221198901198961119909+ 11986233119890minus1198962119909+ 119862441198901198962119909

+ 11986255119890minus(119901120592)119909

(38)

where the parameters of 119862119894119894(119894 = 1 2 3 4) are dependent on

119901Substituting 119906 from (32) and 120579 from (38) into (26) the

following relationships become clear

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198621

11986222=

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198622

11986233= minus

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198623

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198624

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(39)

To obtain the parameters of 119862119894(119894 = 1 2 3 4) and 119862

119894119894(119894 =

1 2 3 4) (32) and (38) are substituted into the equation ofboundary conditions as follows

1198621+ 1198622+ 1198623+ 1198624= minus1198625

1198621119890minus1198961119897+ 11986221198901198961119897+ 1198623119890minus1198962119897+ 11986241198901198962119897= minus1198625119890minus(119901120592)119897

minus 119862111198961+ 119862221198961minus 119862331198962+ 119862441198962= (

119901

120592

)11986255

minus 119862111198961119890minus1198961119897+ 1198622211989611198901198961119897minus 119862331198962119890minus1198962119897+ 1198624411989621198901198962119897

= (

119901

120592

)11986255119890minus(119901120592)119897

(40)

Solving (40) we obtain 119862119894(119894 = 1 2 3 4) as follows

1198621=

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198622= minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198623= minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

1198624=

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

(41)

Substituting (41) into (32) we obtain

119906 =

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625119890minus1198961119909

minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

11986251198901198961119909

minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625119890minus1198962119909

+

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

11986251198901198962119909

+ 1198625119890minus(119901120592)119909

(42)

The relationship between 119862119894and 119862

119894119894in (39) gives

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986222

= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986233=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(43)

Substituting (43) into (38) we obtain

120579 = minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

6 Advances in Materials Science and Engineering

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909+

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625119890minus(119901120592)119909

(44)

Substituting (42) and (44) into (25) yields

120590 = minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909minus 1205732120592 (119901 + 120576) 119862

5119890minus(119901120592)119909

(45)

4 Numerical Inversion of the Transforms

In order to determine the nondimensional temperaturenondimensional displacement and nondimensional stressin the rod we need to invert the parameters of 120579 119906 and120590 from the Laplace domain Unfortunately the obtainedsolutions in the Laplace domain are too complicated tobe inverted analytically thus a feasible numerical methodthat is the Riemann-sum approximation method is used tocomplete the inversion Accordingly any function 119891(119909 119901) inthe Laplace domain can be inverted to the time domain asfollows [48]

119891 (119909 119905)

=

119890120573119905

119905

[

1

2

119891 (119909 120573) + Re119873

sum

119899=1

119891(119909 120573 +

119894119899120587

119905

) (minus1)119899]

(46)

where Re is the real part and 119894 is the imaginary number unitFor faster convergence numerous numerical experimentshave shown that the value of 120573 satisfies the relation 120573119905 asymp 47

[48]

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 1 Nondimensional temperature distribution for different 119905values

5 Numerical Results and Discussions

In terms of the Riemann-sum approximation defined in (46)numerical Laplace inversion is implemented to obtain thenondimensional temperature displacement and stress in therod in the time domain For the purposes of simulationthe thermoelastic material is specified as copper and theparameters are

120582 = 776 times 1010Nmminus2

120583 = 386 times 1010Nmminus2

120588 = 8954 kgmminus3

120572119905= 178 times 10

minus5 Kminus1

119862119864= 3831 JKgminus1 Kminus1

120581 = 386Wmminus1 Kminus1

1198790= 293K

120577 = 00005Kminus1

(47)

The other constants are

1198760= 10

1205910= 005

119897 = 10

(48)

Numerical calculation is carried out for the following fivecases

In Case 1 we investigate the nondimensional tempera-ture displacement and stress varying with time as shown inFigures 1ndash3 with the moving heat source velocity fractionalorder parameter value of temperature-dependent properties

Advances in Materials Science and Engineering 7

1 2 3 4 5 6 7 8 9 10minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 2 Nondimensional stress distribution for different 119905 values

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 3 Nondimensional displacement distribution for different 119905values

and magnetic field set as constants (ie 120592 = 20 120572 = 025120599 = 05 120576 = 5) In Case 2 the considered variables varyingwith different moving heat source velocity are investigatedwhen 119905 = 15 120572 = 025 120599 = 05 and 120576 = 5 (Figures 4ndash6) In Case 3 we investigate how the considered variablesvary with different magnetic fields when 119905 = 15 120592 = 20120599 = 05 and 120572 = 025 (Figures 7ndash9) Case 4 involvesthe considered variables varying with different temperature-dependent properties when 119905 = 15 120592 = 20 120572 = 025 and120576 = 5 (Figures 10ndash12) and in Case 5 we investigate howthe considered variables vary with different fractional orderparameters when 119905 = 15 120592 = 20 120599 = 05 and 120576 = 5 (Figures13ndash15)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 4 Nondimensional temperature distribution for different 120592values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00N

ondi

men

siona

l stre

ss 120590

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 5 Nondimensional stress distribution for different 120592 values

In Figures 1ndash3 (Case 1) the solid line dash line and dotline refer to 119905 = 15 119905 = 20 and 119905 = 25 respectively Figure 1shows that the nondimensional temperature increases as thetime increases At location 119909 = 120592119905 the heat source releases itsmaximum energy which leads to a peak value As shown inFigure 2 the nondimensional stress in the rod is compressiveDue to the fixed ends thermal expansion deformation isrestrained in both ends and leads to the occurrence ofcompressive thermal stress in the rod The absolute valueof stress increases with the passage of time As shown inFigure 3 the nondimensional displacement also increasesas time progresses Due to the applied moving heat sourcethe rod undergoes thermal expansion deformation With thepassage of time the heat disturbed region enlarges so thatthermal expansion deformation evolves along the rod

8 Advances in Materials Science and Engineering

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 6 Nondimensional displacement distribution for different 120592values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 7 Nondimensional temperature distribution for different 120576values

In Figures 4ndash6 (Case 2) the solid line dash line anddot line refer to 120592 = 20 120592 = 30 and 120592 = 40 respec-tively As shown in Figure 4 the nondimensional temper-ature decreases with the increasing of the moving heat re-source velocity In a given period the energy that the heatsource can release is constant however the intensity ofthe released energy per unit length decreases as the sourcespeed increases It is also clear in Figures 5 and 6 that themagnitudes of the nondimensional stress and displacementdecrease as the moving heat source velocity increases whichis the result of reduction in heat energy intensity per unitlength at greater velocities

In Figures 7ndash9 (Case 3) the solid line dash line and dotline refer to 120576 = 0 120576 = 50 and 120576 = 100 respectively

1 2 3 4 5 6 7 8 9 10minus07

minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 8 Nondimensional stress distribution for different 120576 values

0 1 2 3 4 5 6 7 8 9 100000

0005

0010

0015

0020

0025

0030N

ondi

men

siona

l disp

lace

men

t u

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 9 Nondimensional displacement distribution for different 120576values

In Figure 7 the three lines overlap because the magneticfield has no effect on the temperature As shown in Figure 8the absolute value of stress increases as the 120576 value increasesprior to the point where the curves intersect then afterthe intersection the absolute value of stress decreases as theapplied magnetic field value increases Figure 9 for Case 3shows that nondimensional displacement decreases as themagnetic field increases The largest displacement value is inthe case of 120576 = 0 namely in the absence of the magnetic fieldThis indicates that themagnetic field acts to damp the thermalexpansion deformation of the rod

In Figures 10ndash12 (Case 4) the solid line dash line anddot line refer to 120599 = 05 120599 = 075 and 120599 = 10 respectivelyAs shown in Figure 10 the temperature increases as the 120599

Advances in Materials Science and Engineering 9

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 10 Nondimensional temperature distribution for different 120599values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 11 Nondimensional stress distribution for different 120599 values

value increases before the three curves intersect at whichpoint it decreases as the temperature-dependent propertiesincrease Figures 11 and 12 show where the displacementand absolute value of stress decrease as the temperature-dependent properties increase

Figures 13ndash15 (Case 5) show the variations of the nondi-mensional temperature stress and displacement respec-tively which demonstrate the effects of the fractional orderparameter on the variations of the considered variables Tothis end a series of values of the fractional order parameter120572 within the region (0 1] are tested through numericalcalculation As representation of the effect of 120572 three typicalvalues of 120572 120572 = 025 120572 = 05 and 120572 = 10 are consideredFigures 13ndash15 show that temperature displacement and

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 12 Nondimensional displacement distribution for different120599 values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 13 Nondimensional temperature distribution for different 120572values

absolute stress value all decrease as the fractional order valueincreases prior to the intersection of the three curves Afterthe intersection however all considered variables increasewith the increasing of the fractional order parameter

6 Conclusions

A generalized magnetothermoelastic rod with temperature-dependent properties subjected to a moving heat sourceis investigated in the fractional order theory The resultsprovided the following most notable conclusions

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

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Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

4 Advances in Materials Science and Engineering

In terms of the nondimensional variables in (18) (14) takethe following forms (dropping the asterisks for convenience)

120599120590 = 1205732 120597119906

120597119909

minus 119887120579

1205972119906

1205971199092minus

119887

1205732

120597120579

120597119909

= 120599(

1205972119906

1205971199052+ 120576

120597119906

120597119905

)

1205972120579

1205971199092= (1 +

120591120572

0

120572

120597120572

120597119905120572)(120599

120597120579

120597119905

+ 119892

1205972119906

120597119909120597119905

minus 119876)

(19)

where

120599 =

1

1 minus 1205771198790

1205732=

120582 + 2120583

120583

119887 =

1205741198790

120583

119892 =

120574

120599120588119862119864

120576 =

12059001205832

01198672

0

1205780(120582 + 2120583)

(20)

The rod is assumed to have reference temperature 1198790and

homogeneous initial conditions

119906 (119909 0) = (119909 0) = 0

120579 (119909 0) =120579 (119909 0) = 0

(21)

The rod is fixed at both ends with a nondimensional length 119897so the boundary conditions are

119906 (0 119905) = 119906 (119897 119905) = 0

120597120579 (0 119905)

120597119909

=

120597120579 (119897 119905)

120597119909

= 0

(22)

The heat source is moving along the 119909-axis with a constantvelocity 120592 which can be described as follows

119876 = 1198760120575 (119909 minus 120592119905) (23)

where 1198760is a constant and 120575 is the delta function

Applying the Laplace transform defined by

119871 [119891 (119905)] = 119891 (119901) = int

infin

0

119890minus119901119905119891 (119905) 119889119905 Re (119901) gt 0 (24)

to (19) with (21) we obtain

120599120590 = 1205732 119889119906

119889119909

minus 119887120579 (25)

1198892119906

1198891199092minus

119887

1205732

119889120579

119889119909

= 120599119901 (119901 + 120576) 119906 (26)

1198892120579

1198891199092= (1 +

120591120572

0

120572

119901120572)(119901120599120579 + 119892119901

119889119906

119889119909

minus 120596120599119890minus(119901120592)119909

) (27)

where

120596 =

1198760

120592

(28)

By applying the Laplace transform the boundary conditionsin (21) can be expressed as follows

119906 (0 119901) = 119906 (119897 119901) = 0

119889120579 (0 119901)

119889119909

=

119889120579 (119897 119901)

119889119909

= 0

(29)

3 Solutions in the Laplace Domain

Eliminating 120579 between (26) and (27) we obtain the followingequation satisfied by 119906

1198894119906

1198891199094minus 1198981

1198892119906

1198891199092+ 1198982119906 = 119898

3119890minus(119901120592)119909

(30)

where

1198981= (120599 +

119892119887

1205732)(1 +

120591120572

0

120572

119901120572)119901 + 120599119901 (119901 + 120576)

1198982= (1 +

120591120572

0

120572

119901120572)12059921199012(119901 + 120576)

1198983=

119887120596119901120599 (1 + (120591120572

0120572) 119901

120572)

1205732120592

(31)

The general solution of (30) is

119906 = 1198621119890minus1198961119909+ 11986221198901198961119909+ 1198623119890minus1198962119909+ 11986241198901198962119909

+ 1198625119890minus(119901120592)119909

(32)

where 119862119894(119894 = 1 2 3 4) are parameters depending on 119901 to be

determined from the boundary conditions and

1198625=

1198983

[(119901120592)4minus 1198981(119901120592)

2+ 1198982]

(33)

where 1198961and 119896

2are the roots of the following characteristic

equation

1198964minus 11989811198962+ 1198982= 0 (34)

and 1198961 1198962are given by

1198961=radic1198981+ radic119898

12minus 41198982

2

1198962=radic1198981minus radic119898

12minus 41198982

2

(35)

Similarly eliminating 119906 between (26) and (27) we obtain

1198894120579

1198891199094minus 1198981

1198892120579

1198891199092+ 1198982120579 = 119898

4119890minus(119901120592)119909

(36)

Advances in Materials Science and Engineering 5

where

1198984= 120596120599(1 +

120591120572

0

120572

119901120572)[120599119901 (119901 + 120576) minus

1199012

1205922] (37)

The general solution of (36) is

120579 = 11986211119890minus1198961119909+ 119862221198901198961119909+ 11986233119890minus1198962119909+ 119862441198901198962119909

+ 11986255119890minus(119901120592)119909

(38)

where the parameters of 119862119894119894(119894 = 1 2 3 4) are dependent on

119901Substituting 119906 from (32) and 120579 from (38) into (26) the

following relationships become clear

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198621

11986222=

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198622

11986233= minus

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198623

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198624

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(39)

To obtain the parameters of 119862119894(119894 = 1 2 3 4) and 119862

119894119894(119894 =

1 2 3 4) (32) and (38) are substituted into the equation ofboundary conditions as follows

1198621+ 1198622+ 1198623+ 1198624= minus1198625

1198621119890minus1198961119897+ 11986221198901198961119897+ 1198623119890minus1198962119897+ 11986241198901198962119897= minus1198625119890minus(119901120592)119897

minus 119862111198961+ 119862221198961minus 119862331198962+ 119862441198962= (

119901

120592

)11986255

minus 119862111198961119890minus1198961119897+ 1198622211989611198901198961119897minus 119862331198962119890minus1198962119897+ 1198624411989621198901198962119897

= (

119901

120592

)11986255119890minus(119901120592)119897

(40)

Solving (40) we obtain 119862119894(119894 = 1 2 3 4) as follows

1198621=

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198622= minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198623= minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

1198624=

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

(41)

Substituting (41) into (32) we obtain

119906 =

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625119890minus1198961119909

minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

11986251198901198961119909

minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625119890minus1198962119909

+

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

11986251198901198962119909

+ 1198625119890minus(119901120592)119909

(42)

The relationship between 119862119894and 119862

119894119894in (39) gives

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986222

= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986233=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(43)

Substituting (43) into (38) we obtain

120579 = minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

6 Advances in Materials Science and Engineering

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909+

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625119890minus(119901120592)119909

(44)

Substituting (42) and (44) into (25) yields

120590 = minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909minus 1205732120592 (119901 + 120576) 119862

5119890minus(119901120592)119909

(45)

4 Numerical Inversion of the Transforms

In order to determine the nondimensional temperaturenondimensional displacement and nondimensional stressin the rod we need to invert the parameters of 120579 119906 and120590 from the Laplace domain Unfortunately the obtainedsolutions in the Laplace domain are too complicated tobe inverted analytically thus a feasible numerical methodthat is the Riemann-sum approximation method is used tocomplete the inversion Accordingly any function 119891(119909 119901) inthe Laplace domain can be inverted to the time domain asfollows [48]

119891 (119909 119905)

=

119890120573119905

119905

[

1

2

119891 (119909 120573) + Re119873

sum

119899=1

119891(119909 120573 +

119894119899120587

119905

) (minus1)119899]

(46)

where Re is the real part and 119894 is the imaginary number unitFor faster convergence numerous numerical experimentshave shown that the value of 120573 satisfies the relation 120573119905 asymp 47

[48]

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 1 Nondimensional temperature distribution for different 119905values

5 Numerical Results and Discussions

In terms of the Riemann-sum approximation defined in (46)numerical Laplace inversion is implemented to obtain thenondimensional temperature displacement and stress in therod in the time domain For the purposes of simulationthe thermoelastic material is specified as copper and theparameters are

120582 = 776 times 1010Nmminus2

120583 = 386 times 1010Nmminus2

120588 = 8954 kgmminus3

120572119905= 178 times 10

minus5 Kminus1

119862119864= 3831 JKgminus1 Kminus1

120581 = 386Wmminus1 Kminus1

1198790= 293K

120577 = 00005Kminus1

(47)

The other constants are

1198760= 10

1205910= 005

119897 = 10

(48)

Numerical calculation is carried out for the following fivecases

In Case 1 we investigate the nondimensional tempera-ture displacement and stress varying with time as shown inFigures 1ndash3 with the moving heat source velocity fractionalorder parameter value of temperature-dependent properties

Advances in Materials Science and Engineering 7

1 2 3 4 5 6 7 8 9 10minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 2 Nondimensional stress distribution for different 119905 values

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 3 Nondimensional displacement distribution for different 119905values

and magnetic field set as constants (ie 120592 = 20 120572 = 025120599 = 05 120576 = 5) In Case 2 the considered variables varyingwith different moving heat source velocity are investigatedwhen 119905 = 15 120572 = 025 120599 = 05 and 120576 = 5 (Figures 4ndash6) In Case 3 we investigate how the considered variablesvary with different magnetic fields when 119905 = 15 120592 = 20120599 = 05 and 120572 = 025 (Figures 7ndash9) Case 4 involvesthe considered variables varying with different temperature-dependent properties when 119905 = 15 120592 = 20 120572 = 025 and120576 = 5 (Figures 10ndash12) and in Case 5 we investigate howthe considered variables vary with different fractional orderparameters when 119905 = 15 120592 = 20 120599 = 05 and 120576 = 5 (Figures13ndash15)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 4 Nondimensional temperature distribution for different 120592values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00N

ondi

men

siona

l stre

ss 120590

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 5 Nondimensional stress distribution for different 120592 values

In Figures 1ndash3 (Case 1) the solid line dash line and dotline refer to 119905 = 15 119905 = 20 and 119905 = 25 respectively Figure 1shows that the nondimensional temperature increases as thetime increases At location 119909 = 120592119905 the heat source releases itsmaximum energy which leads to a peak value As shown inFigure 2 the nondimensional stress in the rod is compressiveDue to the fixed ends thermal expansion deformation isrestrained in both ends and leads to the occurrence ofcompressive thermal stress in the rod The absolute valueof stress increases with the passage of time As shown inFigure 3 the nondimensional displacement also increasesas time progresses Due to the applied moving heat sourcethe rod undergoes thermal expansion deformation With thepassage of time the heat disturbed region enlarges so thatthermal expansion deformation evolves along the rod

8 Advances in Materials Science and Engineering

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 6 Nondimensional displacement distribution for different 120592values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 7 Nondimensional temperature distribution for different 120576values

In Figures 4ndash6 (Case 2) the solid line dash line anddot line refer to 120592 = 20 120592 = 30 and 120592 = 40 respec-tively As shown in Figure 4 the nondimensional temper-ature decreases with the increasing of the moving heat re-source velocity In a given period the energy that the heatsource can release is constant however the intensity ofthe released energy per unit length decreases as the sourcespeed increases It is also clear in Figures 5 and 6 that themagnitudes of the nondimensional stress and displacementdecrease as the moving heat source velocity increases whichis the result of reduction in heat energy intensity per unitlength at greater velocities

In Figures 7ndash9 (Case 3) the solid line dash line and dotline refer to 120576 = 0 120576 = 50 and 120576 = 100 respectively

1 2 3 4 5 6 7 8 9 10minus07

minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 8 Nondimensional stress distribution for different 120576 values

0 1 2 3 4 5 6 7 8 9 100000

0005

0010

0015

0020

0025

0030N

ondi

men

siona

l disp

lace

men

t u

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 9 Nondimensional displacement distribution for different 120576values

In Figure 7 the three lines overlap because the magneticfield has no effect on the temperature As shown in Figure 8the absolute value of stress increases as the 120576 value increasesprior to the point where the curves intersect then afterthe intersection the absolute value of stress decreases as theapplied magnetic field value increases Figure 9 for Case 3shows that nondimensional displacement decreases as themagnetic field increases The largest displacement value is inthe case of 120576 = 0 namely in the absence of the magnetic fieldThis indicates that themagnetic field acts to damp the thermalexpansion deformation of the rod

In Figures 10ndash12 (Case 4) the solid line dash line anddot line refer to 120599 = 05 120599 = 075 and 120599 = 10 respectivelyAs shown in Figure 10 the temperature increases as the 120599

Advances in Materials Science and Engineering 9

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 10 Nondimensional temperature distribution for different 120599values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 11 Nondimensional stress distribution for different 120599 values

value increases before the three curves intersect at whichpoint it decreases as the temperature-dependent propertiesincrease Figures 11 and 12 show where the displacementand absolute value of stress decrease as the temperature-dependent properties increase

Figures 13ndash15 (Case 5) show the variations of the nondi-mensional temperature stress and displacement respec-tively which demonstrate the effects of the fractional orderparameter on the variations of the considered variables Tothis end a series of values of the fractional order parameter120572 within the region (0 1] are tested through numericalcalculation As representation of the effect of 120572 three typicalvalues of 120572 120572 = 025 120572 = 05 and 120572 = 10 are consideredFigures 13ndash15 show that temperature displacement and

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 12 Nondimensional displacement distribution for different120599 values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 13 Nondimensional temperature distribution for different 120572values

absolute stress value all decrease as the fractional order valueincreases prior to the intersection of the three curves Afterthe intersection however all considered variables increasewith the increasing of the fractional order parameter

6 Conclusions

A generalized magnetothermoelastic rod with temperature-dependent properties subjected to a moving heat sourceis investigated in the fractional order theory The resultsprovided the following most notable conclusions

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 5

where

1198984= 120596120599(1 +

120591120572

0

120572

119901120572)[120599119901 (119901 + 120576) minus

1199012

1205922] (37)

The general solution of (36) is

120579 = 11986211119890minus1198961119909+ 119862221198901198961119909+ 11986233119890minus1198962119909+ 119862441198901198962119909

+ 11986255119890minus(119901120592)119909

(38)

where the parameters of 119862119894119894(119894 = 1 2 3 4) are dependent on

119901Substituting 119906 from (32) and 120579 from (38) into (26) the

following relationships become clear

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198621

11986222=

1205732[1198961

2minus 120599119901 (119901 + 120576)]

1198871198961

1198622

11986233= minus

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198623

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)]

1198871198962

1198624

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(39)

To obtain the parameters of 119862119894(119894 = 1 2 3 4) and 119862

119894119894(119894 =

1 2 3 4) (32) and (38) are substituted into the equation ofboundary conditions as follows

1198621+ 1198622+ 1198623+ 1198624= minus1198625

1198621119890minus1198961119897+ 11986221198901198961119897+ 1198623119890minus1198962119897+ 11986241198901198962119897= minus1198625119890minus(119901120592)119897

minus 119862111198961+ 119862221198961minus 119862331198962+ 119862441198962= (

119901

120592

)11986255

minus 119862111198961119890minus1198961119897+ 1198622211989611198901198961119897minus 119862331198962119890minus1198962119897+ 1198624411989621198901198962119897

= (

119901

120592

)11986255119890minus(119901120592)119897

(40)

Solving (40) we obtain 119862119894(119894 = 1 2 3 4) as follows

1198621=

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198622= minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625

1198623= minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

1198624=

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625

(41)

Substituting (41) into (32) we obtain

119906 =

(1198962

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

1198625119890minus1198961119909

minus

(1198962

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

11986251198901198961119909

minus

(1198961

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

1198625119890minus1198962119909

+

(1198961

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

11986251198901198962119909

+ 1198625119890minus(119901120592)119909

(42)

The relationship between 119862119894and 119862

119894119894in (39) gives

11986211= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986222

= minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625

11986233=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986244=

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625

11986255=

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625

(43)

Substituting (43) into (38) we obtain

120579 = minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732[1198961

2minus 120599119901 (119901 + 120576)] (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198871198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

6 Advances in Materials Science and Engineering

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909+

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625119890minus(119901120592)119909

(44)

Substituting (42) and (44) into (25) yields

120590 = minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909minus 1205732120592 (119901 + 120576) 119862

5119890minus(119901120592)119909

(45)

4 Numerical Inversion of the Transforms

In order to determine the nondimensional temperaturenondimensional displacement and nondimensional stressin the rod we need to invert the parameters of 120579 119906 and120590 from the Laplace domain Unfortunately the obtainedsolutions in the Laplace domain are too complicated tobe inverted analytically thus a feasible numerical methodthat is the Riemann-sum approximation method is used tocomplete the inversion Accordingly any function 119891(119909 119901) inthe Laplace domain can be inverted to the time domain asfollows [48]

119891 (119909 119905)

=

119890120573119905

119905

[

1

2

119891 (119909 120573) + Re119873

sum

119899=1

119891(119909 120573 +

119894119899120587

119905

) (minus1)119899]

(46)

where Re is the real part and 119894 is the imaginary number unitFor faster convergence numerous numerical experimentshave shown that the value of 120573 satisfies the relation 120573119905 asymp 47

[48]

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 1 Nondimensional temperature distribution for different 119905values

5 Numerical Results and Discussions

In terms of the Riemann-sum approximation defined in (46)numerical Laplace inversion is implemented to obtain thenondimensional temperature displacement and stress in therod in the time domain For the purposes of simulationthe thermoelastic material is specified as copper and theparameters are

120582 = 776 times 1010Nmminus2

120583 = 386 times 1010Nmminus2

120588 = 8954 kgmminus3

120572119905= 178 times 10

minus5 Kminus1

119862119864= 3831 JKgminus1 Kminus1

120581 = 386Wmminus1 Kminus1

1198790= 293K

120577 = 00005Kminus1

(47)

The other constants are

1198760= 10

1205910= 005

119897 = 10

(48)

Numerical calculation is carried out for the following fivecases

In Case 1 we investigate the nondimensional tempera-ture displacement and stress varying with time as shown inFigures 1ndash3 with the moving heat source velocity fractionalorder parameter value of temperature-dependent properties

Advances in Materials Science and Engineering 7

1 2 3 4 5 6 7 8 9 10minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 2 Nondimensional stress distribution for different 119905 values

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 3 Nondimensional displacement distribution for different 119905values

and magnetic field set as constants (ie 120592 = 20 120572 = 025120599 = 05 120576 = 5) In Case 2 the considered variables varyingwith different moving heat source velocity are investigatedwhen 119905 = 15 120572 = 025 120599 = 05 and 120576 = 5 (Figures 4ndash6) In Case 3 we investigate how the considered variablesvary with different magnetic fields when 119905 = 15 120592 = 20120599 = 05 and 120572 = 025 (Figures 7ndash9) Case 4 involvesthe considered variables varying with different temperature-dependent properties when 119905 = 15 120592 = 20 120572 = 025 and120576 = 5 (Figures 10ndash12) and in Case 5 we investigate howthe considered variables vary with different fractional orderparameters when 119905 = 15 120592 = 20 120599 = 05 and 120576 = 5 (Figures13ndash15)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 4 Nondimensional temperature distribution for different 120592values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00N

ondi

men

siona

l stre

ss 120590

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 5 Nondimensional stress distribution for different 120592 values

In Figures 1ndash3 (Case 1) the solid line dash line and dotline refer to 119905 = 15 119905 = 20 and 119905 = 25 respectively Figure 1shows that the nondimensional temperature increases as thetime increases At location 119909 = 120592119905 the heat source releases itsmaximum energy which leads to a peak value As shown inFigure 2 the nondimensional stress in the rod is compressiveDue to the fixed ends thermal expansion deformation isrestrained in both ends and leads to the occurrence ofcompressive thermal stress in the rod The absolute valueof stress increases with the passage of time As shown inFigure 3 the nondimensional displacement also increasesas time progresses Due to the applied moving heat sourcethe rod undergoes thermal expansion deformation With thepassage of time the heat disturbed region enlarges so thatthermal expansion deformation evolves along the rod

8 Advances in Materials Science and Engineering

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 6 Nondimensional displacement distribution for different 120592values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 7 Nondimensional temperature distribution for different 120576values

In Figures 4ndash6 (Case 2) the solid line dash line anddot line refer to 120592 = 20 120592 = 30 and 120592 = 40 respec-tively As shown in Figure 4 the nondimensional temper-ature decreases with the increasing of the moving heat re-source velocity In a given period the energy that the heatsource can release is constant however the intensity ofthe released energy per unit length decreases as the sourcespeed increases It is also clear in Figures 5 and 6 that themagnitudes of the nondimensional stress and displacementdecrease as the moving heat source velocity increases whichis the result of reduction in heat energy intensity per unitlength at greater velocities

In Figures 7ndash9 (Case 3) the solid line dash line and dotline refer to 120576 = 0 120576 = 50 and 120576 = 100 respectively

1 2 3 4 5 6 7 8 9 10minus07

minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 8 Nondimensional stress distribution for different 120576 values

0 1 2 3 4 5 6 7 8 9 100000

0005

0010

0015

0020

0025

0030N

ondi

men

siona

l disp

lace

men

t u

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 9 Nondimensional displacement distribution for different 120576values

In Figure 7 the three lines overlap because the magneticfield has no effect on the temperature As shown in Figure 8the absolute value of stress increases as the 120576 value increasesprior to the point where the curves intersect then afterthe intersection the absolute value of stress decreases as theapplied magnetic field value increases Figure 9 for Case 3shows that nondimensional displacement decreases as themagnetic field increases The largest displacement value is inthe case of 120576 = 0 namely in the absence of the magnetic fieldThis indicates that themagnetic field acts to damp the thermalexpansion deformation of the rod

In Figures 10ndash12 (Case 4) the solid line dash line anddot line refer to 120599 = 05 120599 = 075 and 120599 = 10 respectivelyAs shown in Figure 10 the temperature increases as the 120599

Advances in Materials Science and Engineering 9

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 10 Nondimensional temperature distribution for different 120599values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 11 Nondimensional stress distribution for different 120599 values

value increases before the three curves intersect at whichpoint it decreases as the temperature-dependent propertiesincrease Figures 11 and 12 show where the displacementand absolute value of stress decrease as the temperature-dependent properties increase

Figures 13ndash15 (Case 5) show the variations of the nondi-mensional temperature stress and displacement respec-tively which demonstrate the effects of the fractional orderparameter on the variations of the considered variables Tothis end a series of values of the fractional order parameter120572 within the region (0 1] are tested through numericalcalculation As representation of the effect of 120572 three typicalvalues of 120572 120572 = 025 120572 = 05 and 120572 = 10 are consideredFigures 13ndash15 show that temperature displacement and

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 12 Nondimensional displacement distribution for different120599 values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 13 Nondimensional temperature distribution for different 120572values

absolute stress value all decrease as the fractional order valueincreases prior to the intersection of the three curves Afterthe intersection however all considered variables increasewith the increasing of the fractional order parameter

6 Conclusions

A generalized magnetothermoelastic rod with temperature-dependent properties subjected to a moving heat sourceis investigated in the fractional order theory The resultsprovided the following most notable conclusions

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

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NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 Advances in Materials Science and Engineering

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732[1198962

2minus 120599119901 (119901 + 120576)] (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198871198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909+

1205732[1205922120599 (119901 + 120576) minus 119901]

119887120592

1198625119890minus(119901120592)119909

(44)

Substituting (42) and (44) into (25) yields

120590 = minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (1198901198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 1198625119890minus1198961119909

minus

1205732119901 (119901 + 120576) (119896

2

2minus 11990121205922) (119890minus1198961119897minus 119890minus(119901120592)119897

)

1198961(1198961

2minus 1198962

2) (1198901198961119897minus 119890minus1198961119897)

sdot 11986251198901198961119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (1198901198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 1198625119890minus1198962119909

+

1205732119901 (119901 + 120576) (119896

1

2minus 11990121205922) (119890minus1198962119897minus 119890minus(119901120592)119897

)

1198962(1198961

2minus 1198962

2) (1198901198962119897minus 119890minus1198962119897)

sdot 11986251198901198962119909minus 1205732120592 (119901 + 120576) 119862

5119890minus(119901120592)119909

(45)

4 Numerical Inversion of the Transforms

In order to determine the nondimensional temperaturenondimensional displacement and nondimensional stressin the rod we need to invert the parameters of 120579 119906 and120590 from the Laplace domain Unfortunately the obtainedsolutions in the Laplace domain are too complicated tobe inverted analytically thus a feasible numerical methodthat is the Riemann-sum approximation method is used tocomplete the inversion Accordingly any function 119891(119909 119901) inthe Laplace domain can be inverted to the time domain asfollows [48]

119891 (119909 119905)

=

119890120573119905

119905

[

1

2

119891 (119909 120573) + Re119873

sum

119899=1

119891(119909 120573 +

119894119899120587

119905

) (minus1)119899]

(46)

where Re is the real part and 119894 is the imaginary number unitFor faster convergence numerous numerical experimentshave shown that the value of 120573 satisfies the relation 120573119905 asymp 47

[48]

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 1 Nondimensional temperature distribution for different 119905values

5 Numerical Results and Discussions

In terms of the Riemann-sum approximation defined in (46)numerical Laplace inversion is implemented to obtain thenondimensional temperature displacement and stress in therod in the time domain For the purposes of simulationthe thermoelastic material is specified as copper and theparameters are

120582 = 776 times 1010Nmminus2

120583 = 386 times 1010Nmminus2

120588 = 8954 kgmminus3

120572119905= 178 times 10

minus5 Kminus1

119862119864= 3831 JKgminus1 Kminus1

120581 = 386Wmminus1 Kminus1

1198790= 293K

120577 = 00005Kminus1

(47)

The other constants are

1198760= 10

1205910= 005

119897 = 10

(48)

Numerical calculation is carried out for the following fivecases

In Case 1 we investigate the nondimensional tempera-ture displacement and stress varying with time as shown inFigures 1ndash3 with the moving heat source velocity fractionalorder parameter value of temperature-dependent properties

Advances in Materials Science and Engineering 7

1 2 3 4 5 6 7 8 9 10minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 2 Nondimensional stress distribution for different 119905 values

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 3 Nondimensional displacement distribution for different 119905values

and magnetic field set as constants (ie 120592 = 20 120572 = 025120599 = 05 120576 = 5) In Case 2 the considered variables varyingwith different moving heat source velocity are investigatedwhen 119905 = 15 120572 = 025 120599 = 05 and 120576 = 5 (Figures 4ndash6) In Case 3 we investigate how the considered variablesvary with different magnetic fields when 119905 = 15 120592 = 20120599 = 05 and 120572 = 025 (Figures 7ndash9) Case 4 involvesthe considered variables varying with different temperature-dependent properties when 119905 = 15 120592 = 20 120572 = 025 and120576 = 5 (Figures 10ndash12) and in Case 5 we investigate howthe considered variables vary with different fractional orderparameters when 119905 = 15 120592 = 20 120599 = 05 and 120576 = 5 (Figures13ndash15)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 4 Nondimensional temperature distribution for different 120592values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00N

ondi

men

siona

l stre

ss 120590

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 5 Nondimensional stress distribution for different 120592 values

In Figures 1ndash3 (Case 1) the solid line dash line and dotline refer to 119905 = 15 119905 = 20 and 119905 = 25 respectively Figure 1shows that the nondimensional temperature increases as thetime increases At location 119909 = 120592119905 the heat source releases itsmaximum energy which leads to a peak value As shown inFigure 2 the nondimensional stress in the rod is compressiveDue to the fixed ends thermal expansion deformation isrestrained in both ends and leads to the occurrence ofcompressive thermal stress in the rod The absolute valueof stress increases with the passage of time As shown inFigure 3 the nondimensional displacement also increasesas time progresses Due to the applied moving heat sourcethe rod undergoes thermal expansion deformation With thepassage of time the heat disturbed region enlarges so thatthermal expansion deformation evolves along the rod

8 Advances in Materials Science and Engineering

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 6 Nondimensional displacement distribution for different 120592values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 7 Nondimensional temperature distribution for different 120576values

In Figures 4ndash6 (Case 2) the solid line dash line anddot line refer to 120592 = 20 120592 = 30 and 120592 = 40 respec-tively As shown in Figure 4 the nondimensional temper-ature decreases with the increasing of the moving heat re-source velocity In a given period the energy that the heatsource can release is constant however the intensity ofthe released energy per unit length decreases as the sourcespeed increases It is also clear in Figures 5 and 6 that themagnitudes of the nondimensional stress and displacementdecrease as the moving heat source velocity increases whichis the result of reduction in heat energy intensity per unitlength at greater velocities

In Figures 7ndash9 (Case 3) the solid line dash line and dotline refer to 120576 = 0 120576 = 50 and 120576 = 100 respectively

1 2 3 4 5 6 7 8 9 10minus07

minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 8 Nondimensional stress distribution for different 120576 values

0 1 2 3 4 5 6 7 8 9 100000

0005

0010

0015

0020

0025

0030N

ondi

men

siona

l disp

lace

men

t u

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 9 Nondimensional displacement distribution for different 120576values

In Figure 7 the three lines overlap because the magneticfield has no effect on the temperature As shown in Figure 8the absolute value of stress increases as the 120576 value increasesprior to the point where the curves intersect then afterthe intersection the absolute value of stress decreases as theapplied magnetic field value increases Figure 9 for Case 3shows that nondimensional displacement decreases as themagnetic field increases The largest displacement value is inthe case of 120576 = 0 namely in the absence of the magnetic fieldThis indicates that themagnetic field acts to damp the thermalexpansion deformation of the rod

In Figures 10ndash12 (Case 4) the solid line dash line anddot line refer to 120599 = 05 120599 = 075 and 120599 = 10 respectivelyAs shown in Figure 10 the temperature increases as the 120599

Advances in Materials Science and Engineering 9

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 10 Nondimensional temperature distribution for different 120599values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 11 Nondimensional stress distribution for different 120599 values

value increases before the three curves intersect at whichpoint it decreases as the temperature-dependent propertiesincrease Figures 11 and 12 show where the displacementand absolute value of stress decrease as the temperature-dependent properties increase

Figures 13ndash15 (Case 5) show the variations of the nondi-mensional temperature stress and displacement respec-tively which demonstrate the effects of the fractional orderparameter on the variations of the considered variables Tothis end a series of values of the fractional order parameter120572 within the region (0 1] are tested through numericalcalculation As representation of the effect of 120572 three typicalvalues of 120572 120572 = 025 120572 = 05 and 120572 = 10 are consideredFigures 13ndash15 show that temperature displacement and

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 12 Nondimensional displacement distribution for different120599 values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 13 Nondimensional temperature distribution for different 120572values

absolute stress value all decrease as the fractional order valueincreases prior to the intersection of the three curves Afterthe intersection however all considered variables increasewith the increasing of the fractional order parameter

6 Conclusions

A generalized magnetothermoelastic rod with temperature-dependent properties subjected to a moving heat sourceis investigated in the fractional order theory The resultsprovided the following most notable conclusions

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 7

1 2 3 4 5 6 7 8 9 10minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 2 Nondimensional stress distribution for different 119905 values

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

t = 15

t = 20

t = 25

Figure 3 Nondimensional displacement distribution for different 119905values

and magnetic field set as constants (ie 120592 = 20 120572 = 025120599 = 05 120576 = 5) In Case 2 the considered variables varyingwith different moving heat source velocity are investigatedwhen 119905 = 15 120572 = 025 120599 = 05 and 120576 = 5 (Figures 4ndash6) In Case 3 we investigate how the considered variablesvary with different magnetic fields when 119905 = 15 120592 = 20120599 = 05 and 120572 = 025 (Figures 7ndash9) Case 4 involvesthe considered variables varying with different temperature-dependent properties when 119905 = 15 120592 = 20 120572 = 025 and120576 = 5 (Figures 10ndash12) and in Case 5 we investigate howthe considered variables vary with different fractional orderparameters when 119905 = 15 120592 = 20 120599 = 05 and 120576 = 5 (Figures13ndash15)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 4 Nondimensional temperature distribution for different 120592values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00N

ondi

men

siona

l stre

ss 120590

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 5 Nondimensional stress distribution for different 120592 values

In Figures 1ndash3 (Case 1) the solid line dash line and dotline refer to 119905 = 15 119905 = 20 and 119905 = 25 respectively Figure 1shows that the nondimensional temperature increases as thetime increases At location 119909 = 120592119905 the heat source releases itsmaximum energy which leads to a peak value As shown inFigure 2 the nondimensional stress in the rod is compressiveDue to the fixed ends thermal expansion deformation isrestrained in both ends and leads to the occurrence ofcompressive thermal stress in the rod The absolute valueof stress increases with the passage of time As shown inFigure 3 the nondimensional displacement also increasesas time progresses Due to the applied moving heat sourcethe rod undergoes thermal expansion deformation With thepassage of time the heat disturbed region enlarges so thatthermal expansion deformation evolves along the rod

8 Advances in Materials Science and Engineering

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 6 Nondimensional displacement distribution for different 120592values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 7 Nondimensional temperature distribution for different 120576values

In Figures 4ndash6 (Case 2) the solid line dash line anddot line refer to 120592 = 20 120592 = 30 and 120592 = 40 respec-tively As shown in Figure 4 the nondimensional temper-ature decreases with the increasing of the moving heat re-source velocity In a given period the energy that the heatsource can release is constant however the intensity ofthe released energy per unit length decreases as the sourcespeed increases It is also clear in Figures 5 and 6 that themagnitudes of the nondimensional stress and displacementdecrease as the moving heat source velocity increases whichis the result of reduction in heat energy intensity per unitlength at greater velocities

In Figures 7ndash9 (Case 3) the solid line dash line and dotline refer to 120576 = 0 120576 = 50 and 120576 = 100 respectively

1 2 3 4 5 6 7 8 9 10minus07

minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 8 Nondimensional stress distribution for different 120576 values

0 1 2 3 4 5 6 7 8 9 100000

0005

0010

0015

0020

0025

0030N

ondi

men

siona

l disp

lace

men

t u

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 9 Nondimensional displacement distribution for different 120576values

In Figure 7 the three lines overlap because the magneticfield has no effect on the temperature As shown in Figure 8the absolute value of stress increases as the 120576 value increasesprior to the point where the curves intersect then afterthe intersection the absolute value of stress decreases as theapplied magnetic field value increases Figure 9 for Case 3shows that nondimensional displacement decreases as themagnetic field increases The largest displacement value is inthe case of 120576 = 0 namely in the absence of the magnetic fieldThis indicates that themagnetic field acts to damp the thermalexpansion deformation of the rod

In Figures 10ndash12 (Case 4) the solid line dash line anddot line refer to 120599 = 05 120599 = 075 and 120599 = 10 respectivelyAs shown in Figure 10 the temperature increases as the 120599

Advances in Materials Science and Engineering 9

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 10 Nondimensional temperature distribution for different 120599values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 11 Nondimensional stress distribution for different 120599 values

value increases before the three curves intersect at whichpoint it decreases as the temperature-dependent propertiesincrease Figures 11 and 12 show where the displacementand absolute value of stress decrease as the temperature-dependent properties increase

Figures 13ndash15 (Case 5) show the variations of the nondi-mensional temperature stress and displacement respec-tively which demonstrate the effects of the fractional orderparameter on the variations of the considered variables Tothis end a series of values of the fractional order parameter120572 within the region (0 1] are tested through numericalcalculation As representation of the effect of 120572 three typicalvalues of 120572 120572 = 025 120572 = 05 and 120572 = 10 are consideredFigures 13ndash15 show that temperature displacement and

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 12 Nondimensional displacement distribution for different120599 values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 13 Nondimensional temperature distribution for different 120572values

absolute stress value all decrease as the fractional order valueincreases prior to the intersection of the three curves Afterthe intersection however all considered variables increasewith the increasing of the fractional order parameter

6 Conclusions

A generalized magnetothermoelastic rod with temperature-dependent properties subjected to a moving heat sourceis investigated in the fractional order theory The resultsprovided the following most notable conclusions

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 Advances in Materials Science and Engineering

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120592 = 20

120592 = 30

120592 = 40

Figure 6 Nondimensional displacement distribution for different 120592values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 7 Nondimensional temperature distribution for different 120576values

In Figures 4ndash6 (Case 2) the solid line dash line anddot line refer to 120592 = 20 120592 = 30 and 120592 = 40 respec-tively As shown in Figure 4 the nondimensional temper-ature decreases with the increasing of the moving heat re-source velocity In a given period the energy that the heatsource can release is constant however the intensity ofthe released energy per unit length decreases as the sourcespeed increases It is also clear in Figures 5 and 6 that themagnitudes of the nondimensional stress and displacementdecrease as the moving heat source velocity increases whichis the result of reduction in heat energy intensity per unitlength at greater velocities

In Figures 7ndash9 (Case 3) the solid line dash line and dotline refer to 120576 = 0 120576 = 50 and 120576 = 100 respectively

1 2 3 4 5 6 7 8 9 10minus07

minus06

minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 8 Nondimensional stress distribution for different 120576 values

0 1 2 3 4 5 6 7 8 9 100000

0005

0010

0015

0020

0025

0030N

ondi

men

siona

l disp

lace

men

t u

Nondimensional coordinate X

120576 = 0

120576 = 50

120576 = 100

Figure 9 Nondimensional displacement distribution for different 120576values

In Figure 7 the three lines overlap because the magneticfield has no effect on the temperature As shown in Figure 8the absolute value of stress increases as the 120576 value increasesprior to the point where the curves intersect then afterthe intersection the absolute value of stress decreases as theapplied magnetic field value increases Figure 9 for Case 3shows that nondimensional displacement decreases as themagnetic field increases The largest displacement value is inthe case of 120576 = 0 namely in the absence of the magnetic fieldThis indicates that themagnetic field acts to damp the thermalexpansion deformation of the rod

In Figures 10ndash12 (Case 4) the solid line dash line anddot line refer to 120599 = 05 120599 = 075 and 120599 = 10 respectivelyAs shown in Figure 10 the temperature increases as the 120599

Advances in Materials Science and Engineering 9

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 10 Nondimensional temperature distribution for different 120599values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 11 Nondimensional stress distribution for different 120599 values

value increases before the three curves intersect at whichpoint it decreases as the temperature-dependent propertiesincrease Figures 11 and 12 show where the displacementand absolute value of stress decrease as the temperature-dependent properties increase

Figures 13ndash15 (Case 5) show the variations of the nondi-mensional temperature stress and displacement respec-tively which demonstrate the effects of the fractional orderparameter on the variations of the considered variables Tothis end a series of values of the fractional order parameter120572 within the region (0 1] are tested through numericalcalculation As representation of the effect of 120572 three typicalvalues of 120572 120572 = 025 120572 = 05 and 120572 = 10 are consideredFigures 13ndash15 show that temperature displacement and

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 12 Nondimensional displacement distribution for different120599 values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 13 Nondimensional temperature distribution for different 120572values

absolute stress value all decrease as the fractional order valueincreases prior to the intersection of the three curves Afterthe intersection however all considered variables increasewith the increasing of the fractional order parameter

6 Conclusions

A generalized magnetothermoelastic rod with temperature-dependent properties subjected to a moving heat sourceis investigated in the fractional order theory The resultsprovided the following most notable conclusions

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 9

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 10 Nondimensional temperature distribution for different 120599values

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 11 Nondimensional stress distribution for different 120599 values

value increases before the three curves intersect at whichpoint it decreases as the temperature-dependent propertiesincrease Figures 11 and 12 show where the displacementand absolute value of stress decrease as the temperature-dependent properties increase

Figures 13ndash15 (Case 5) show the variations of the nondi-mensional temperature stress and displacement respec-tively which demonstrate the effects of the fractional orderparameter on the variations of the considered variables Tothis end a series of values of the fractional order parameter120572 within the region (0 1] are tested through numericalcalculation As representation of the effect of 120572 three typicalvalues of 120572 120572 = 025 120572 = 05 and 120572 = 10 are consideredFigures 13ndash15 show that temperature displacement and

0 1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120599 = 05

120599 = 075

120599 = 10

Figure 12 Nondimensional displacement distribution for different120599 values

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Non

dim

ensio

nal t

empe

ratu

re 120579

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 13 Nondimensional temperature distribution for different 120572values

absolute stress value all decrease as the fractional order valueincreases prior to the intersection of the three curves Afterthe intersection however all considered variables increasewith the increasing of the fractional order parameter

6 Conclusions

A generalized magnetothermoelastic rod with temperature-dependent properties subjected to a moving heat sourceis investigated in the fractional order theory The resultsprovided the following most notable conclusions

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

10 Advances in Materials Science and Engineering

1 2 3 4 5 6 7 8 9 10minus05

minus04

minus03

minus02

minus01

00

Non

dim

ensio

nal s

tress

120590

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 14 Nondimensional stress distribution for different120572 values

1 2 3 4 5 6 7 8 9 100000

0001

0002

0003

0004

0005

0006

0007

0008

Non

dim

ensio

nal d

ispla

cem

ent u

Nondimensional coordinate X

120572 = 025

120572 = 05

120572 = 10

Figure 15 Nondimensional displacement distribution for different120572 values

(1) The effects of the fractional parameter 120572 on all theconsidered variables are very significant as clearlyevidenced by the peak values of the curves

(2) The magnetic field also has a significant effect onthe solution of the nondimensional displacement andstress but barely influences the variation in nondi-mensional temperature

(3) The temperature-dependent properties play a signifi-cant role on all distributions

As a final remark the results presented in this papershould prove useful for researchers in material sciencesdesigners of new materials low-temperature physics re-searchers and those working to further develop the theoryof thermoelasticity with fractional calculusThe introduction

of fractional calculus and variable temperature-dependentmodulus to the generalized thermoelastic rod may providea more realistic model for these studies

Nomenclature

119869 Current density vector119880 Displacement vector119861 Magnetic induction vector120590119894119895 The components of stress tensor

119890119894119895 The components of strain tensor

119890119896119896 Cubic dilation

119906119894 The components of displacement vector

120579 120579 = 119879 minus 1198790

119879 Absolute temperature of the medium1198790 Reference temperature

120581119894119895 The coefficient of thermal conductivity

1205910 Thermal relaxation time

120588 Mass density119862119864 Specific heat at constant strain

120582 120583 Lamersquos constants120572119905 Linear thermal expansion coefficient

119876 The strength of the applied heat source perunit mass

120574 (3120582 + 2120583)120572119905

120578 The entropy density119902119894 The components of heat flux vector

120592 Velocity of the moving heat source120572 Constant parameter such that 0 lt 120572 le 1

Competing Interests

The authors declare that they have no competing interests

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] C Catteneo ldquoA form of heat conduction equation whicheliminates the paradox of instantaneous propagationrdquoComputeRendus vol 247 pp 431ndash433 1958

[5] P Vernotte ldquoSome possible complications in the phenomenonof thermal conductionrdquo Compute Rendus vol 252 pp 2190ndash2191 1961

[6] H M Youssef ldquoGeneralized thermoelastic infinite mediumwith spherical cavity subjected to moving heat sourcerdquo Compu-tational Mathematics and Modeling vol 21 pp 212ndash225 2010

[7] H H Sherief and N M El-Maghraby ldquoEffect of body forces ona 2D generalized thermoelastic long cylinderrdquo Computers andMathematics with Applications vol 66 no 7 pp 1181ndash1191 2013

[8] M I A Othman and I A Abbas ldquoEffect of rotation onplane waves in generalized thermomicrostretch elastic solidcomparison of different theories using finite element methodrdquoCanadian Journal of Physics vol 92 no 10 pp 1269ndash1277 2014

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 11

[9] H H Sherief and F A Hamza ldquoModeling of variable thermalconductivity in a generalized thermoelastic infinitely longhollow cylinderrdquoMeccanica vol 51 no 3 pp 551ndash558 2016

[10] Y Heydarpour and M Aghdam ldquoTransient analysis of rotatingfunctionally graded truncated conical shells based on the LordndashShulman modelrdquo Thin-Walled Structures vol 104 pp 168ndash1842016

[11] Y Wang D Liu Q Wang and J Zhou ldquoAsymptotic analysisof thermoelastic response in functionally graded thin platesubjected to a transient thermal shockrdquo Composite Structuresvol 139 pp 233ndash242 2016

[12] T He and L Cao ldquoA problem of generalized magneto-thermoelastic thin slim strip subjected to amoving heat sourcerdquoMathematical and Computer Modelling vol 49 no 7-8 pp1710ndash1720 2009

[13] H H Sherief and S E Khader ldquoPropagation of discontinuitiesin electromagneto generalized thermoelasticity in cylindricalregionsrdquoMeccanica vol 48 no 10 pp 2511ndash2523 2013

[14] I A Abbas and AM Zenkour ldquoLS model on electro-magneto-thermoelastic response of an infinite functionally graded cylin-derrdquo Composite Structures vol 96 pp 89ndash96 2013

[15] N Sarkar ldquoGeneralized magneto-thermoelasticity with modi-fiedOhmrsquos law under three theoriesrdquoComputationalMathemat-ics and Modeling vol 25 no 4 pp 544ndash564 2014

[16] P Pal P Das and M Kanoria ldquoMagneto-thermoelasticresponse in a functionally graded rotating medium due to aperiodically varying heat sourcerdquo Acta Mechanica vol 226 no7 pp 2103ndash2120 2015

[17] M C Singh and N Chakraborty ldquoReflection of a planemagneto-thermoelastic wave at the boundary of a solid half-space in presence of initial stressrdquo Applied Mathematical Mod-elling vol 39 no 5-6 pp 1409ndash1421 2015

[18] S M Said ldquoInfluence of gravity on generalized magneto-thermoelastic medium for three-phase-lag modelrdquo Journal ofComputational and Applied Mathematics vol 291 pp 142ndash1572016

[19] M Caputo ldquoVibrations on an infinite viscoelastic layer witha dissipative memoryrdquo Journal of the Acoustical Society ofAmerica vol 56 pp 897ndash904 1974

[20] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[21] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[22] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[23] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2005

[24] H M Youssef ldquoTheory of fractional order generalized ther-moelasticityrdquo Journal of Heat Transfer vol 132 no 6 Article ID061301 7 pages 2010

[25] N Sarkar and A Lahiri ldquoEffect of fractional parameter onplane waves in a rotating elastic medium under fractional ordergeneralized thermoelasticityrdquo International Journal of AppliedMechanics vol 4 no 3 Article ID 1250030 2012

[26] H M Youssef ldquoState-space approach to fractional order two-temperature generalized thermoelastic medium subjected to

moving heat sourcerdquo Mechanics of Advanced Materials andStructures vol 20 no 1 pp 47ndash60 2013

[27] Y J Yu X G Tian and T J Lu ldquoFractional order gener-alized electro-magneto-thermo-elasticityrdquo European Journal ofMechanics A Solids vol 42 pp 188ndash202 2013

[28] H M Youssef and I A Abbas ldquoFractional order generalizedthermoelasticity with variable thermal conductivityrdquo Journal ofVibroengineering vol 16 pp 4077ndash4087 2014

[29] Y Q Song J T Bai Z Zhao and Y F Kang ldquoStudy on thevibration of optically excitedmicrocantilevers under fractional-order thermoelastic theoryrdquo International Journal of Thermo-physics vol 36 no 4 pp 733ndash746 2015

[30] I A Abbas and H M Youssef ldquoTwo-dimensional fractionalorder generalized thermoelastic porous materialrdquo Latin Ameri-can Journal of Solids and Structures vol 12 no 7 pp 1415ndash14312015

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

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

[33] H Sherief and A M Abd El-Latief ldquoEffect of variable ther-mal conductivity on a half-space under the fractional ordertheory of thermoelasticityrdquo International Journal of MechanicalSciences vol 74 pp 185ndash189 2013

[34] THHe andYGuo ldquoAone-dimensional thermoelastic problemdue to a moving heat source under fractional order theory ofthermoelasticityrdquo Advances in Materials Science and Engineer-ing vol 2014 Article ID 510205 9 pages 2014

[35] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 1D problem for a half-spacerdquo ZAMM-Zeitschrift fur Angewandte Mathematik undMechanik vol 94 no 6 pp 509ndash515 2014

[36] HH Sherief andAM Abd El-Latief ldquoApplication of fractionalorder theory of thermoelasticity to a 2D problem for a half-spacerdquo Applied Mathematics and Computation vol 248 pp584ndash592 2014

[37] I A Abbas ldquoEigenvalue approach to fractional order general-ized magneto-thermoelastic medium subjected to moving heatsourcerdquo Journal of Magnetism and Magnetic Materials vol 377pp 452ndash459 2015

[38] H H Sherief and A M Abd El-Latief ldquoA one-dimensionalfractional order thermoelastic problem for a spherical cavityrdquoMathematics and Mechanics of Solids vol 20 no 5 pp 512ndash5212015

[39] Y B Ma and T H He ldquoDynamic response of a generalizedpiezoelectric-thermoelastic problem under fractional ordertheory of thermoelasticityrdquo Mechanics of Advanced Materialsand Structures vol 23 no 10 pp 1173ndash1180 2016

[40] M A Ezzat A S El-Karamany and A A Samaan ldquoThe depen-dence of the modulus of elasticity on reference temperature ingeneralized thermoelasticity with thermal relaxationrdquo AppliedMathematics and Computation vol 147 no 1 pp 169ndash189 2004

[41] M N Allam K A Elsibai and A E Abouelregal ldquoMagneto-thermoelasticity for an infinite body with a spherical cavityand variable material properties without energy dissipationrdquoInternational Journal of Solids and Structures vol 47 no 20 pp2631ndash2638 2010

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

12 Advances in Materials Science and Engineering

[42] Q-L Xiong and X-G Tian ldquoTransient magneto-thermoelasticresponse for a semi-infinite body with voids and variable mate-rial properties during thermal shockrdquo International Journal ofApplied Mechanics vol 3 no 4 pp 881ndash902 2011

[43] A E Abouelregal ldquoFractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heatingrdquoJournal of Thermal Stresses vol 34 no 11 pp 1139ndash1155 2011

[44] M I A Othman Y D Elmaklizi and S M Said ldquoGeneralizedthermoelastic medium with temperature-dependent propertiesfor different theories under the effect of gravity fieldrdquo Interna-tional Journal of Thermophysics vol 34 no 3 pp 521ndash537 2013

[45] P Pal A Kar and M Kanoria ldquoFractional order generalisedthermoelasticity to an infinite body with a cylindrical cavityand variable material propertiesrdquo European Journal of Compu-tational Mechanics vol 23 no 1-2 pp 96ndash111 2014

[46] Q Xiong and X Tian ldquoGeneralized magneto-thermo-microstretch response of a half-space with temperature-dependent properties during thermal shockrdquo Latin AmericanJournal of Solids and Structures vol 12 no 13 pp 2562ndash25802015

[47] Y Z Wang D Liu Q Wang and J Z Zhou ldquoFractional ordertheory of thermoelasticity for elastic medium with variablematerial propertiesrdquo Journal of Thermal Stresses vol 38 no 6pp 665ndash676 2015

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

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials