Research ArticleEffect of Initial Stress on the Propagation Characteristics ofWaves in Fiber-Reinforced Transversely Isotropic ThermoelasticMaterial under an Inviscid Liquid Layer

Rajneesh Kumar1 Sanjeev Ahuja2 and S K Garg3

1 Department of Mathematics Kurukshetra University Kurukshetra Haryana 136119 India2University Institute of Engg amp Tech Kurukshetra University Kurukshetra Haryana 136119 India3 Department of Mathematics Deen Bandhu Chotu Ram Uni of Sc amp Tech Sonipat Haryana 131027 India

Correspondence should be addressed to Sanjeev Ahuja sanjeev ahuja81hotmailcom

Received 24 May 2014 Accepted 17 July 2014 Published 26 August 2014

Academic Editor Felix Sharipov

Copyright copy 2014 Rajneesh Kumar et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The present investigation deals with the propagation of waves in fiber-reinforced transversely isotropic thermoelastic solid halfspace with initial stresses under a layer of inviscid liquidThe secular equation for surface equation in compact form is derived afterdeveloping the mathematical model The phase velocity and attenuation coefficients of plane waves are studied numerically for aparticular model Effects of initial stress and thickness of the layer on the phase velocity attenuation coefficient and specific lossof energy are predicted graphically in the certain model A particular case of Rayleigh wave has been discussed and the dispersioncurves of the phase velocity and attenuation coefficients have also been presented graphically Some other particular cases are alsodeduced from the present investigation

1 Introduction

Fiber-reinforced materials are widely used in engineeringstructures due to their superiority over the structural mate-rials in applications requiring high strength and stiffness inlightweight components Consequently characterization oftheir mechanical behavior is of particular importance forstructural design using these materials Fibers are assumedan inherent material property rather than some form ofinclusion in models as Spencer [1] In the case of an elasticsolid reinforced by a series of parallel fibers it is usual toassume transverse isotropy

The idea of continuous self-reinforcement at every pointof an elastic solid was introduced by Belfield et al [2] Thecharacteristic property of reinforced concrete member isthat its components namely concrete and steel act togetheras a single anisotropic unit as main long as they remainin the elastic condition that is the two components arebound together so that there can be no relative displace-ment between them The dynamical interaction between thethermal and mechanical fields in solids has great practical

applications in modern aeronautics astronautics nuclearreactors and high energy particle accelerators

The analysis of stress and deformation of fiber-reinforcedcomposite materials has been an important subject of solidmechanics for last three decades Pipkin [3] did pioneerworkson the subject Sengupta and Nath [4] discussed the problemof surface waves in fiber-reinforced anisotropic elastic media

Lord and Shulman [5] introduced a theory of generalizedthermoelasticity with one relaxation time for an isotropicbody The theory was extended for anisotropic body byDhaliwal and Sherief [6] In this theory a modified law ofheat conduction including both the heat flux and its timederivatives replaces the conventional Fourierrsquos Law The heatequation associated with this theory is hyperbolic and henceeliminates the paradox of infinite speeds of propagationinherent in both coupled and uncoupled theories of thermoe-lasticity Erdem [7] derived heat conduction equation for acomposite rigid material containing an arbitrary distributionof fibers Recently Kumar and Gupta [8] discussed the Wavemotion in an anisotropic fiber-reinforced thermoelastic solid

Hindawi Publishing CorporationJournal of ermodynamicsVolume 2014 Article ID 134276 10 pageshttpdxdoiorg1011552014134276

2 Journal of Thermodynamics

Surface waves carry a lot of information about the Earthrsquoscrust and dispersion analysis of surface waves is concernedwith the phase velocity andwave number Further the velocityof seismic waves depends upon the elastic parameters ofthe medium through which they travel Therefore manyinvestigators have studied the surface wave propagation byconsidering different kinds of models Chadwick and Seet[9] and Singh and Sharma [10] have discussed the prop-agation of plane harmonic waves in transversely isotropicthermoelasticmaterials Singh [11] studied a problemonwavepropagation in an anisotropic generalized thermoelastic solidand obtained a cubic equation which gives the dimensionalvelocities of various plane waves

Chadwick [12] has discussed the propagation of surfacewaves in homogeneous thermoelastic media Chadwick andWindle [13] studied the effect of heat conduction on thepropagation of Rayleigh waves in the semi-infinite media(i) when the surface is maintained at constant temperatureand (ii) when the surface is thermally insulated Sharma andSingh [14] have studied thermoelastic surface waves in atransversely isotropic half-space

The study of wave propagation in a generalized ther-moelastic media with additional parameters like prestressporosity viscosity microstructure temperature and otherparameters provide vital information about existence of newormodifiedwaves Such informationmay be useful for exper-imental seismologists in correcting earthquake estimation

Prestressedmaterials have various applications for exam-ple in oil and geophysical industry NDT in prestressedmaterials use of rubber composites in automotive andaerospace and defense industries (often in prestressed states)and in the study of biological tissues (lung tendon etc)which are all nonlinear prestressed viscoelastic composites

Montanaro [15] investigated the isotropic linear thermoe-lasticity with hydrostatic initial stress Wang and Slattery[16] formulated the thermoelastic equations without energydissipation for a body which has previously received a largedeformation and is at nonuniform temperature Iesan [17]presented a theory of Cosserat thermoelastic solids withinitial stresses Ames and Straughan [18] derived the continu-ous dependence results for initially prestressed thermoelasticbodies

Some theorems in the generalized theory of thermoe-lasticity for prestressed bodies were studied by Wang etal [19] Marin and Marinescu [20] studied the asymptoticpartition of total energy for the solutions of the mixedinitial boundary value problem within the context of thethermoelasticity of initially stressed bodies Kalinchuk et al[21] studied the problemof steady-state harmonic oscillationsfor a nonhomogeneous thermoelastic prestressed medium

Othman and Song [22] discussed the reflection ofplane waves from a thermoelastic elastic solid half-spaceunder hydrostatic initial stress without energy dissipationSingh [23] studied wave propagation in an initially stressedtransversely isotropic thermoelastic half-space However noattempt has been made to discuss the effect of hydrostaticinitial stress on propagation of waves in the layer of fiber-reinforced transversely isotropic thermoelasticmaterial Oth-man et al [24] investigated the influences of fractional

order hydrostatic initial stress and gravity field on the planewaves in a fiber-reinforced isotropic thermoelastic mediumusing normal mode analysis Abd-Alla et al [25] studiedthe effects of rotation and gravity field on surface wavesin fiber-reinforced thermoelastic media under four theoriesOthman and Atwa [26] studied the effect of rotation gravityand reinforcement on the total deformation and mutualinteraction of the body in a fiber-reinforced thermoelasticsolid using normal mode analysis in the context of Green-Naghdi theory of thermoelasitcity Abbas [27] studied thetransient phenomena in the magnetothermoelastic model inthe context of the Lord and Shulman theory in a perfectlyconducting medium Abbas [28] constructed the equationsfor generalized thermoelasticity of an unbounded fiber-reinforced anisotropic medium with a circular hole in thecontext of Green and Naghdi (GN) theory Recently Abbasand Zenkour [29] investigated two-temperature generalizedthermoelasticity with two relaxation times for an infinitefiber-reinforced anisotropic plate containing a circular cavityusing a finite element method

In the present paper the governing equations of thermoe-lastcity of fiber-reinforced transversely isotropic solid halfspace with initial stresses under a layer of inviscid liquidare formulated and solved analytically in two dimensionsat uniform temperature The phase velocity and attenuationcoefficients of plane waves are studied numerically for aparticular model Specific loss of energy is obtained anddiscussed numerically for a particular model to study theeffects of initial stress The study of present investigation hasapplications in civil engineering and geophysics

2 Basic Equations

The basic equations in the dynamic theory of the homoge-neous thermally conducting fiber-reinforced medium withan initial hydrostatic stress without body forces and heatsources are given by Lord and Shulman [5] and Abbas andOthman [30] as

120590119894119895119895minus 119875120596119894119895119895= 120588119894 119894 119895 = 1 2 3 (1)

and heat conduction equation is given by

119870119894119895119879119894119895= (

120597

120597119905

+ 1205910

1205972

1205971199052)(119879119900120573119894119895119894119895+ 120588119862119890119879) 119894 119895 = 1 2 3

(2)

The constitutive relations for thermally conducting trans-versely isotropic fiber-reinforced linearly elastic media are

120590119894119895= 120582119890119896119896120575119894119895+ 2120583119879119890119894119895+ 120572 (119886

119896119886119898119890119896119898120575119894119895+ 119886119894119886119895119890119896119896)

+ 2 (120583119871minus 120583119879) (119886119894119886119896119890119896119895+ 119886119895119886119896119890119896119894)

+ 120573 (119886119896119886119898119890119896119898119886119894119886119895) minus 120573119894119895119879120575119894119895

119894 119895 119896 119898 = 1 2 3

(3)

Journal of Thermodynamics 3

where

119890119894119895=

1

2

(119906119894119895+ 119906119895119894) 120596

119894119895=

1

2

(119906119895119894minus 119906119894119895)

119894 119895 = 1 2 3

(4)

and 120582 120583119879are elastic parameters 120572 120573 (120583

119871minus120583119879) are reinforced

anisotropic elastic parameters 120588 is the mass density 120590119894119895

are components of stress tensor 119906119894are the displacement

components 119890119894119895are components of infinitesimal strain 119879

is the temperature change of a material particle 119879119900is the

reference uniform temperature of the body 119896119894119895are coefficients

of thermal conductivity 120573119894119895

are thermal elastic couplingtensor 119862

119890is the specific heat at constant strain 120575

119894119895is the

Kronecker delta and 119875 is the initial Pressure The commanotation is used for spatial derivatives and superimposeddot represents time differentiation 119886

119895are components of a

all referred to Cartesian coordinate The vector a may be afunction of position We choose a so that its components are(1 0 0)

The equation which governs the motion in terms ofvelocity potential 120595 of the homogeneous inviscid liquid isgiven by Ewing et al [31]

1205972120595

1205971199092

1

+

1205972120595

1205971199092

2

=

1

(120572ℓ)2

1205972120595

1205971199052 (5)

where 120572ℓ = radic120582ℓ120588ℓ is the velocity of sound liquid

The displacement components 119906ℓ1 119906ℓ

2and pressure 119901ℓ in

the medium1198722are given by

119906ℓ

1=

120597120595

1205971199091

119906ℓ

2=

120597120595

1205971199092

119901ℓ= minus120588ℓ 120597120595

120597119905

(6)

3 Formulation of the Problem

We consider a homogeneous thermally conducting trans-versely isotropic fiber-reinforced medium (119872

1) with an

initial hydrostatic stress lying under a uniform homogeneousinviscid liquid layer medium (119872

2) of thickness 119867 We take

the rectangular Cartesian coordinate system 119874119909111990921199093at any

point on the plane horizontal surface and 1199091-axis in the

direction of wave propagation and 1199092-axis pointing vertically

downward into the half-space so that all particles on a lineparallel to 119909

3-axis are equally displaced therefore all the field

quantities will be independent of 1199093-coordinates Hence the

liquid layer medium 1198722occupies the region minus119867 lt 119909

2lt 0

and 1199092gt 0 is occupied by the half-space medium 119872

1 The

plane 1199092= minus119867 represents the free surface of the liquid and

1199092= 0 is taken as interface between 119872

1and 119872

2medium

(Figure 1)The displacement components for medium119872

1are taken

as

= (1199061 1199062 0) (7)

Homogeneous inviscid liquid layer

Transversely isotropic fiber-reinforced

0

x2

x2

= minusH

H

x1

(medium M2)

thermal half space (medium M1)

Figure 1 Geometry of the problem

With the help of (3) (4) and (7) and (1)-(2) take the form

11986211

12059721199061

1205971199092

1

+ (11986212+ 1198620minus

119875

2

)

12059721199062

12059711990911205971199092

+ (1198620+

119875

2

)

12059721199061

1205971199092

2

minus 12057311

120597119879

1205971199091

= 120588

12059721199061

1205971199052

(8)

11986222

12059721199062

1205971199092

2

+ (11986212+ 1198620minus

119875

2

)

12059721199061

12059711990911205971199092

+ (1198620+

119875

2

)

12059721199062

1205971199092

1

minus 12057322

120597119879

1205971199092

= 120588

12059721199062

1205971199052

(9)

1205972119879

1205971199092

1

+ 119896

1205972119879

1205971199092

2

minus

120588119862119890

11989611

(

120597119879

120597119905

+ 120591119900

1205972119879

1205971199052)

= 120576 [(

12059721199061

1205971199091120597119905

+ 120591119900

12059731199061

12059711990911205971199052) + 120573(

12059721199062

1205971199092120597119905

+ 120591119900

12059731199062

12059711990921205971199052)]

(10)

where

12057311= (11986211+ 11986212) 12057211+ 1198621212057222

12057322= (11986212+ 11986223minus 11986255) 12057211+ 1198622212057222

11986212= 120582 + 120572 119862

11= 120582 + 2120572 + 4120583

119871minus 2120583119879+ 120573

11986222= 11986233= 120582 + 2120583

119879

11986244= 11986266= 2120583119871 119862

55= 2120583119879

11986223= 11986233minus 11986255 119862

119900=

11986244

2

120576 =

11987911990012057311

11989611

119896 =

11989622

11989611

120573 =

12057322

12057311

(11)

and 120582 120572 120573 120583119871 120583119879are material constants 120572

11 12057222

are com-ponents of linear thermal expansion and 120591

119900is thermal

relaxation time

4 Journal of Thermodynamics

To facilitate the solution the following dimensionlessquantities are introduced

(1199091015840

1 1199091015840

2) =

119908lowast

V1

(1199091 1199092) (119906

1015840

1 1199061015840

2) =

120588V1119908lowast

12057311119879119900

(1199061 1199062)

1199051015840

119894119895=

119905119894119895

12057311119879119900

1198791015840=

119879

119879119900

1199051015840= 119908lowast119905

ℎ1015840=

V1

119908lowastℎ 120591

1015840

119900= 119908lowast120591119900 120595

1015840=

119908lowast120595

V21

(119906ℓ1015840

1 119906ℓ1015840

2) =

119908lowast

V1

(119906ℓ

1 119906ℓ

2) 119901

ℓ1015840

=

119901ℓ

120582ℓ 119905

ℓ1015840

22=

119905ℓ

22

120582ℓ

(12)

where 119908lowast = 1198621198901198621111989611 V21= 11986211120588 and 119908lowast is the character-

istic frequency of the medium

4 Solution of the Problem

We assume the solutions of the form

(1199061 1199062 119879 120595) = (1119882 119877 119878) 119880 exp 119894120585 (119909

1+ 1198981199092minus 119888119905) (13)

where 119888 = 120596120585 is the nondimensional phase velocity 120596 isthe frequency and 120585 is the wave number 119898 is still unknownparameter 1119882 119877 119878 are respectively the amplitude ratios of1199061 1199062 119879 and 119906ℓ

2with respect to 119906

1 Substituting the values of

1199061 1199062and 119879 from (13) in (8) (9) and (10) we obtain

[

119898

11986211

(11986212+ 119862119900minus

119875

2

)]119882 + (

minus1

119894120585

)119877

+ [1 minus 1198882+

1

11986211

(119862119900+

119875

2

)1198982] = 0

[

11986222

11986211

1198982+

1

11986211

(119862119900+

119875

2

) minus 1198882]119882

+ (

minus119898120573

119894120585

)119877 + [

1

11986211

(11986212+ 119862119900minus

119875

2

)119898] = 0

[1205761120573 (minus119898119888 + 119898119888

2120591119900119894120585)]119882 + [1 + 119896119898

2+ (

119888

119894120585

minus 1205911199001198882)]119877

+ 1205761[119888 minus 120591

1199001198882119894120585] = 0

(14)

where

1205761=

1198791199001205732

11

120588119908lowast11989611

(15)

The system of (14) has a nontrivial solution if the deter-minant of the coefficients [1119882 119877]119879 vanishes which yields tothe following polynomial characteristic equation

1198986+ 119889211198984+ 119889311198982+ 11988941= 0 (16)

where

1198891= [119896 (119871

21198921)]

1198892= [minus 119892

11198651198712+ 119896 (119871

11198921+ 11987121198922minus 1198602

1)

+ 11987121198921+ (120573)

2

12057611198651198712]

1198893= [minus 119865119892

1(1198711+ 1205761) minus 119865119892

21198712

+ 1205731198651198601(1 minus 120576

1) + 119896 (119871

11198922) + 11987111198921

+ 11987121198922+ (120573)

2

12057611198651198711+ 1198602

1(119865 minus 1)]

1198894= [minus119892

2119865 (1198711+ 1205761) + 11987111198922]

11988921=

1198892

1198891

11988931=

1198893

1198891

11988941=

1198894

1198891

119865 =

minus119888 + 1205911199001198882119894120585

119894120585

119860 = 1198981198601

1198601=

1

11986211

(11986212+ 119862119900minus

119875

2

)

119866 = 11989211198982+ 1198922 119892

1=

11986222

11986211

1198922=

1

11986211

(119862119900+

119875

2

) minus 1198882

119871 = 1198711+ 11987121198982 119871

1= 1 minus 119888

2

1198712=

1

11986211

(119862119900+

119875

2

)

(17)

The characteristic equation (16) is cubic in1198982 and hencepossesses three roots (1198982

119901 119901 = 1 2 3) Therefore there exist

three types of quasi-waves in transversely isotropic elastichalf-space namely quasi-longitudinal waves (QP) quasi-transverse waves (QSV) and quasi-thermal waves (QT)

The formal expression for displacements and volumefraction field satisfying the radiation condition that Re(119898

119901)ge

0 can be written as

(1199061 1199062 119879) =

3

sum

119901=1

119866119901(1 1198991119901 1198992119901) exp 119894120585 (119909

1+ 1198941198981199011199092minus 119888119905)

(18)

where

1198991119901=

Δ1

Δ

1198992119901=

Δ2

Δ

Δ1= (

minus1205731198981205761119888 (1 minus 120591

119900119888119894120585)

119894120585

minus 119860119884)

Δ2= (119866120576

1119888 (1 minus 120591

119900119888119894120585) minus 119860119885)

Journal of Thermodynamics 5

Δ = (119866119884 +

120573119898119885

119894120585

)

119884 = 1 + 1198961198982+ (

119888

119894120585

minus 1205911199001198882)

119885 = 1205761120573119898119888 (minus1 + 119888120591

119900119894120585)

(19)

and 1198661 1198662 1198663are arbitrary constants

Substituting the values of 120595 from (13) in (5) and aftersimplification we obtain

120595 = [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)] exp (119894120585 (119909

1minus 119888119905))

(20)

where

1198984= plusmn((

V1119888

120572ℓ)

2

minus 1)

12

(21)

and 1198664 1198665are arbitrary constants

The normal displacement and normal stress for medium1198722 with the help of (5) (6) (12) and (20) are given by

119906ℓ

2= 1198984120585 [1198664cos (120585119898

41199092) minus 1198665sin (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

119905ℓ

22= minus 120585

2(1 + 119898

2

4) [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

(22)

Also from (3) we have

11990522= 1198621211990611+ 1198622211990622minus 12057322119879

11990521= 119862119900(11990612+ 11990621)

(23)

5 Boundary Conditions

The appropriate boundary conditions are as follows

(i) vanishing of normal stress component at the freesurface of the liquid layer 119909

2= minus119867 that is

(119905ℓ

22)1198722

= minus119901ℓ= 0 (24)

(ii) continuity of normal stress components at the inter-face 119909

2= 0 that is

(11990522)1198721

= minus119901ℓ (25)

(iii) normal displacement components at the interface1199092= 0 that is

(119906ℓ

2)1198722

= (2)1198721

(26)

(iv) vanishing shear stress for medium1198721at the interface

1199092= 0 that is

(11990521)1198721

= 0 (27)

(v) thermal boundary conditions

120597119879

1205971199092

+ ℎ119879 = 0 (28)

where ℎ 997888rarr 0 corresponds Insulated boundarycondition ℎ 997888rarr infin corresponds Isolated boundarycondition

6 Derivations of the Secular Equations

Substituting the value of 1199062 119879 119906ℓ

2 119905ℓ

22from (18) and (22) in

(24)ndash(28) andwith the aid of (7) and (12) after simplificationwe obtain

1198882= (

120572ℓ

V1

)

2

[

[

(tanminus1 (11987911))

2

+ 12058521198672

12058521198672

]

]

(29)

where

11987911=

119862111198984(119864111198631minus 119864121198632+ 119864131198633)

119894119888120588ℓV21(119899111198631minus 119899121198632+ 119899131198633)

1198641119901= 119894120585 (

11986212

11986211

) + (

11986222

11986211

)1198991119901(1198942120585119898119901) minus (

12057322

12057311

)1198992119901

1198642119901= 120585 (119899

1119901119894 minus 119898119901) 119864

3119901= 1198992119901(ℎ minus 120585119898

119901)

(119901 = 1 2 3)

1198631= 1198642211986433minus 1198643211986423 119863

2= 1198642111986433minus 1198643111986423

1198633= 1198642111986432minus 1198643111986422

(30)

Equation (29) is surface waves secular equation of fiber-reinforced transversely isotropic thermoelastic half-spaceunder a uniform homogeneous inviscid liquid layer Thesecular equation has complete information about the phasevelocity wave number and attenuation coefficient of surfacewaves in such medium

61 Particular Cases

(i) If we take ℎ rarr 0 in (29) we obtain the frequencyequation for the thermally insulated boundary

(ii) If we take ℎ rarr infin in (29) we obtain the frequencyequation for the thermally isolated boundary

(iii) Taking 119875 rarr 0 in secular equation (29) yield thefrequency equation for fiber-reinforced transverselyisotropic thermoelastic half space without initialstress under a uniform homogeneous inviscid liquidlayer

(iv) For 120583119879= 120583119871= 120583 119870

11= 11987022= 119870 120573

11= 12057322= 120573 =

(3120582 + 2120583)120572119905 (29) reduces to the frequency equation

in a homogeneous Isotropic half space under inviscidliquid

(v) 120591119900rarr 0 in (10) we obtained the frequency equation

in fiber-reinforced coupled thermoelastic half spaceunder inviscid liquid layer

6 Journal of Thermodynamics

7 Rayleigh Wave

(1) Taking119867 rarr 0 the frequency equation (29) reduces to

11986411(1198642211986433minus 1198642311986432) minus 11986412(1198642111986433minus 1198643111986423)

+ 11986413(1198642111986432minus 1198643111986422) = 0

(31)

Equation (31) is the frequency equation for Rayleighsurface wave propagation in a fiber-reinforced transverselyisotropic thermoelastic half-space with initial stress In gen-eral wave number and hence phase velocity of waves iscomplex quantity therefore the waves are attenuated in spaceIf we write

119888minus1= Vminus1 + 119894120596minus1119865 (32)

so that 120585 = 119870 + 119894119865 where119870 = 120596V V and 119865 are real Also theroots of characteristic equation (16) are in general complexand hence we assume that119898

119901= 119901119901+119894119902119901 so that the exponent

in the plane wave solutions (18) for the half-space becomes

119894119870 (1199091minus 119898119868

1199011199092minus V119905) minus 119870(

119865

119870

1199091+ 119898119877

1199011199092) (33)

where

119898119877

119901= 119901119901minus 119902119901

119865

119870

119898119868

119901= 119902119901+ 119901119901

119865

119870

(119901 = 1 2 3)

(34)

This shows that V is the propagation velocity and 119865 is theattenuation coefficient of wave

The equation (18) can be rewritten as

1199061=

3

sum

119901=1

119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

1199062=

3

sum

119901=1

1198991119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

119879 =

3

sum

119901=1

1198992119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

(35)

with 120582119901= 119870(119898

119877

119901+ 119894119898119868

119901) = 120582

119877

119901+ 119894120582119868

119901(say) (119901 = 1 2 3)

Moreover it is clear that

10038161003816100381610038161003816120582119877

119901

10038161003816100381610038161003816

2

minus

10038161003816100381610038161003816120582119868

119901

10038161003816100381610038161003816

2

= 1198702[(119898119877

119901)

2

minus (119898119868

119901)

2

] (36)

71 Normal Stress Normal stress at the interface 1199092= 0 is

given by

11990522= (119862

12119894120585

3

sum

119901=1

119866119901minus 11986222120585

3

sum

119901=1

1198981199011198661199011198991119901minus 12057322

3

sum

119901=1

1198661199011198992119901)

times exp (119894120585 (1199091minus V119905)) (119901 = 1 2 3)

(37)

72 Specific Loss The Specific loss is the ratio of energy (Δ119882)dissipated in taking a specimen through a stress cycle to theelastic energy (119882) stored in the specimen when the strainis maximum The Specific loss is the most direct method ofdefining internal friction for amaterial For a sinusoidal planewave of small amplitude Kolsky [32] shows that the specificloss Δ119882119882 equals 4120587 times the absolute value of imaginarypart of 120585 to the real part of 120585 that is

Δ119882

119882

= 4120587

10038161003816100381610038161003816100381610038161003816

Im (120585)

Re (120585)

10038161003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

V119866120596

1003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

119866

119865

1003816100381610038161003816100381610038161003816

(38)

8 Numerical Results and Discussion

For numerical computations we take the following valuesof the relevant parameters for generalized fiber-reinforcedtransversely isotropic thermoelastic solid

120582 = 566 times 1010Nm2 120588 = 2660Kgm3

120583119879= 246 times 10

10Nm2 120583119871= 566 times 10

10Nm2

120572 = minus128 times 1010Nm2 120573 = 22090 times 10

10Nm2

11987011= 00921 times 10

3 Jmminus1 degminus1 sminus1 119875 = 100

11987022= 00963 times 10

3 Jmminus1 degminus1 sminus1

1205721= 0017 times 10

4 degminus1

1205722= 0015 times 10

4 degminus1 119879119900= 293K

120591119900= 005 s ℎ = 1 119867 = 1

119862119890= 0787 times 10

3 JKgminus1degminus1 120596 = 2 sminus1(39)

The physical constants for water are given by Ewing et al[31]

120588ℓ= 10 times 10

3 Kgm3 120582ℓ= 214 times 10

3Nm2 (40)

For comparison with the generalized fiber-reinforcedisotropic thermoelastic solid following Ezzat [33] we take

Journal of Thermodynamics 7

0 1 2 3 4 5

Phas

e velo

city

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

25

20

15

10

05

00

times101

Figure 2 Variation of phase velocity wrt wave number (with andwithout initial stress)

the following values of relevant parameters for the case ofcopper material as

120582 = 776 times 1010Nm2

120583119879= 120583119871= 120583 = 386 times 10

10Nm2

11987011= 11987022= 119870 = 386NKminus1 sminus1

12057311= 12057322= 120573 = (3120582 + 2120583) 120572

119905

120572119905= 178 times 10

minus5 Kminus1

1205721= 1005 degminus1 120572

2= 1001 degminus1

119862119890= 3831m2 Kminus1 sminus1 119875 = 0

119867 = 1 120588 = 8954Kgm3

120591119900= 01 s ℎ = 1

(41)

Using the above values of parameters the dispersioncurves of nondimensional phase velocity attenuation coef-ficient and specific heat are shown graphically with respectto nondimensional wave number Here we consider abbre-viations FTTIWIS FTTIIS FTIWIS and FTIIS for fiber-reinforced transversely isotropic thermoelasticmaterial with-out initial stress fiber-reinforced transversely isotropic ther-moelastic material with initial stress fiber-reinforced iso-tropic thermoelastic material without initial stress and fiber-reinforced isotropic thermoelastic material with initial stressrespectively and119867 refers to the thickness of the layer

Figure 2 depicts the variations of phase velocity withrespect to the wave number The behavior of phase velocity

0 1 2 3 4 5Wave number

05

00

minus05

minus10

minus15

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times10minus1

Figure 3 Variation of attenuation coefficient wrt wave number(with and without initial stress)

for FTTIWIS FTTIIS FTIWIS and FTIIS is similar butthe corresponding values are different in magnitude Thevalues of phase velocity decrease for lower wave numberand become stable for higher wave number in all the casesFigure 3 shows the variation of attenuation coefficient withrespect to the wave number For FTTIWIS and FTTIIS thevalues of attenuation coefficient increase monotonically upto wave number (120585) = 2 and after that it becomes constant forwhole range of wave number On the other hand the values ofattenuation coefficients are consistent within the whole rangeof wave number for FTIWIS and FTIIS

Figure 4 indicates the variation of phase velocity withrespect to thewave number for different values of thickness oflayer Here we consider the particular case of fiber-reinforcedtransversely isotropic thermoelastic material and analyses ofthe behavior of curves with respect to initial stress It isobserved that for small value of wave number the graph ofphase velocity shows opposite behavior wrt thickness thatis on increasing the values of thickness of layer the values ofcorresponding phase velocity decreases and it became steadyfor large values of wave number

Figure 5 shows the variation of attenuation coefficientwith respect to the wave number for different value ofthickness of the layer Here also we considered the particularcase of fiber-reinforced transversely isotropic thermoelasticmaterial and analyses of the behavior of curveswith respect toinitial stress It is perceived that on decreasing the value of119867the curve shows consistency for small value of wave numberand on increasing the value of119867 the curve shows uniformityfor large value of 120585

Figure 6 depicts the variations of specific loss verseswave number For FTIWIS and FTIIS the curve shows sharpdecrease in the value of specific loss for the range 0 le 120585 le 1and become parallel for 120585 gt 1 It is observed that graphs

8 Journal of Thermodynamics

0 1 2 3 4 5Wave number

25

20

15

10

05

00

Phas

e velo

city

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

times102

Figure 4 Variation of phase velocity wrt wave number (with andwithout initial stress for different value of119867)

corresponding to FTTIWIS and FTTIIS follow the similarpattern for whole range of 120585 but the values are different inmagnitude

81 Rayleigh Wave Figure 7 shows the variations of phasevelocity of Rayleigh wave verses wave number for FTTIWISFTTIIS FTIWIS and FTIIS The value of phase velocitydecreases for small values of 120585 and become stable for highervalue of 120585 It is analyzed that in all the cases behavior of curvesare almost the same within the range 2 le 120585 le 5 Figure 8shows the variations of attenuation coefficient of Rayleighwave verses wave number for FTTIWIS FTTIIS FTIWISand FTIIS For FTIWIS and FTIIS the graph shows a suddenfall for 120585 = 05 and then increases monotonically up to 120585 = 2and becomes constant for the remaining range of 120585 A largedifference between the behavior of curves for the transverselyisotropic and isotropic case is also perceived

9 Conclusions

The propagation of surface waves in an initially stressedfiber-reinforced transversely isotropic thermoelastic mate-rial under an inviscid liquid layer has been studied Afterdeveloping the formal solutions the secular equation forsurface wave propagation is derived Some special casesof frequency equation are also discussed The attenuationcoefficients and specific loss (liquid layer and half-space)have also been studied and computed for FTTIWIS FTTIISFTIWIS and FTIISThe trend of phase velocity is the same inall these four cases for surface wave but varies for attenuation

0 1 2 3 4 5Wave number

00

05

times10minus1

minus15

minus05

minus10

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

Figure 5 Variation of attenuation coeff wrt wave number (withand without initial stress for different value of119867)

0 1 2 3 4 5

Spec

ific l

oss

40

35

30

25

20

15

10

05

00

minus05

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

times10minus4

Figure 6 Variation of specific loss wrt wave number (with andwithout initial stress)

coefficient From numerical analysis it is shown that thephase speeds and attenuation coefficients for Rayleigh waveare also affected significantly due to the presence of initialstress parameters

Journal of Thermodynamics 9

0 1 2 3 4 5

Phas

e velo

city

Wave number

12

10

80

times103

60

40

20

00

FTTIWISFTTIIS

FTIWISFTIIS

Figure 7 Variation of phase velocity wrt wave number forRayleigh wave (with and without initial stress)

0 1 2 3 4 5

00

minus20

minus40

minus60

minus80

minus10

Wave number

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times105

Figure 8 Variation of attenuation coeff wrt wave number forRayleigh wave (with and without initial stress)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A J M Spencer Deformation of Fibre-reinforced MaterialsUniversity of Oxford Clarendon Va USA 1941

[2] A J Belfield T G Rogers and A JM Spencer ldquoStress in elasticplates reinforced by fibres lying in concentric circlesrdquo Journal oftheMechanics and Physics of Solids vol 31 no 1 pp 25ndash54 1983

[3] A C Pipkin ldquoFinite deformations of ideal fiber-reinforcedcompositesrdquo in Composites Materials G P Sendeckyj Ed vol2 of Mechanics of Composite Materials pp 251ndash308 AcademicPress New York NY USA 1973

[4] P R Sengupta and S Nath ldquoSurface waves in fibre-reinforcedanisotropic elastic mediardquo Indian Academy of Sciences SadhanaProceedings vol 26 pp 363ndash370 2001

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

[6] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 198081

[7] A U Erdem ldquoHeat Conduction in fiber-reinforced rigid bod-iesrdquo 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK) 6ndash8Eylul Ankara Turkey 1995

[8] R Kumar and R R Gupta ldquoStudy of wave motion in ananisotropic fiber-reinforced thermoelastic solidrdquo Journal ofSolid Mechanics vol 2 no 1 pp 91ndash100 2010

[9] P Chadwick and L T C Seet ldquoWave propagation in atransversely isotropic heat-conducting elastic materialrdquo Math-ematika vol 17 pp 255ndash274 1970

[10] H Singh and J N Sharma ldquoGeneralized thermoelastic waves intransversely isotropicmediardquo Journal of the Acoustical Society ofAmerica vol 77 pp 1046ndash1053 1985

[11] B Singh ldquoWave propagation in an anisotropic generalizedthermoelastic solidrdquo Indian Journal of Pure and Applied Mathe-matics vol 34 no 10 pp 1479ndash1485 2003

[12] P Chadwick Progress in Solid Mechanics vol 1 North-HollandAmsterdam The Netherlands 1960 edited by R Hill I NSneddon

[13] P Chadwick and DWWindle ldquoPropagation of Rayleigh wavesalong isothermal and insulated boundariesrdquo Proceedings of theRoyal Society of America vol 280 pp 47ndash71 1964

[14] J N Sharma and H Singh ldquoThermoelastic surface waves ina transversely isotropic half space with thermal relaxationsrdquoIndian Journal of Pure and Applied Mathematics vol 16 no 10pp 1202ndash1219 1985

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

[16] J Wang and P Slattery ldquoThermoelasticity without energydissipation for initially stressed bodiesrdquo International Journal ofMathematics and Mathematical Sciences vol 31 no 6 pp 329ndash337 2002

[17] D Iesan ldquoA theory of prestressed thermoelastic Cosserat con-tinuardquo Journal of Applied Mathematics and Mechanics vol 88no 4 pp 306ndash319 2008

[18] K Ames and B Straughan ldquoContinuous dependence results forinitially prestressed thermoelastic bodiesrdquo International Journalof Engineering Science vol 30 no 1 pp 7ndash13 1992

[19] J Wang R S Dhaliwal and S R Majumdar ldquoSome theoremsin the generalized theory of thermoelasticity for prestressedbodiesrdquo Indian Journal of Pure and Applied Mathematics vol28 no 2 pp 267ndash276 1997

[20] M Marin and C Marinescu ldquoThermoelasticity of initiallystressed bodies asymptotic equipartition of energiesrdquo Interna-tional Journal of Engineering Science vol 36 no 1 pp 73ndash861998

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

2 Journal of Thermodynamics

Surface waves carry a lot of information about the Earthrsquoscrust and dispersion analysis of surface waves is concernedwith the phase velocity andwave number Further the velocityof seismic waves depends upon the elastic parameters ofthe medium through which they travel Therefore manyinvestigators have studied the surface wave propagation byconsidering different kinds of models Chadwick and Seet[9] and Singh and Sharma [10] have discussed the prop-agation of plane harmonic waves in transversely isotropicthermoelasticmaterials Singh [11] studied a problemonwavepropagation in an anisotropic generalized thermoelastic solidand obtained a cubic equation which gives the dimensionalvelocities of various plane waves

Chadwick [12] has discussed the propagation of surfacewaves in homogeneous thermoelastic media Chadwick andWindle [13] studied the effect of heat conduction on thepropagation of Rayleigh waves in the semi-infinite media(i) when the surface is maintained at constant temperatureand (ii) when the surface is thermally insulated Sharma andSingh [14] have studied thermoelastic surface waves in atransversely isotropic half-space

The study of wave propagation in a generalized ther-moelastic media with additional parameters like prestressporosity viscosity microstructure temperature and otherparameters provide vital information about existence of newormodifiedwaves Such informationmay be useful for exper-imental seismologists in correcting earthquake estimation

Prestressedmaterials have various applications for exam-ple in oil and geophysical industry NDT in prestressedmaterials use of rubber composites in automotive andaerospace and defense industries (often in prestressed states)and in the study of biological tissues (lung tendon etc)which are all nonlinear prestressed viscoelastic composites

Montanaro [15] investigated the isotropic linear thermoe-lasticity with hydrostatic initial stress Wang and Slattery[16] formulated the thermoelastic equations without energydissipation for a body which has previously received a largedeformation and is at nonuniform temperature Iesan [17]presented a theory of Cosserat thermoelastic solids withinitial stresses Ames and Straughan [18] derived the continu-ous dependence results for initially prestressed thermoelasticbodies

Some theorems in the generalized theory of thermoe-lasticity for prestressed bodies were studied by Wang etal [19] Marin and Marinescu [20] studied the asymptoticpartition of total energy for the solutions of the mixedinitial boundary value problem within the context of thethermoelasticity of initially stressed bodies Kalinchuk et al[21] studied the problemof steady-state harmonic oscillationsfor a nonhomogeneous thermoelastic prestressed medium

Othman and Song [22] discussed the reflection ofplane waves from a thermoelastic elastic solid half-spaceunder hydrostatic initial stress without energy dissipationSingh [23] studied wave propagation in an initially stressedtransversely isotropic thermoelastic half-space However noattempt has been made to discuss the effect of hydrostaticinitial stress on propagation of waves in the layer of fiber-reinforced transversely isotropic thermoelasticmaterial Oth-man et al [24] investigated the influences of fractional

order hydrostatic initial stress and gravity field on the planewaves in a fiber-reinforced isotropic thermoelastic mediumusing normal mode analysis Abd-Alla et al [25] studiedthe effects of rotation and gravity field on surface wavesin fiber-reinforced thermoelastic media under four theoriesOthman and Atwa [26] studied the effect of rotation gravityand reinforcement on the total deformation and mutualinteraction of the body in a fiber-reinforced thermoelasticsolid using normal mode analysis in the context of Green-Naghdi theory of thermoelasitcity Abbas [27] studied thetransient phenomena in the magnetothermoelastic model inthe context of the Lord and Shulman theory in a perfectlyconducting medium Abbas [28] constructed the equationsfor generalized thermoelasticity of an unbounded fiber-reinforced anisotropic medium with a circular hole in thecontext of Green and Naghdi (GN) theory Recently Abbasand Zenkour [29] investigated two-temperature generalizedthermoelasticity with two relaxation times for an infinitefiber-reinforced anisotropic plate containing a circular cavityusing a finite element method

In the present paper the governing equations of thermoe-lastcity of fiber-reinforced transversely isotropic solid halfspace with initial stresses under a layer of inviscid liquidare formulated and solved analytically in two dimensionsat uniform temperature The phase velocity and attenuationcoefficients of plane waves are studied numerically for aparticular model Specific loss of energy is obtained anddiscussed numerically for a particular model to study theeffects of initial stress The study of present investigation hasapplications in civil engineering and geophysics

2 Basic Equations

The basic equations in the dynamic theory of the homoge-neous thermally conducting fiber-reinforced medium withan initial hydrostatic stress without body forces and heatsources are given by Lord and Shulman [5] and Abbas andOthman [30] as

120590119894119895119895minus 119875120596119894119895119895= 120588119894 119894 119895 = 1 2 3 (1)

and heat conduction equation is given by

119870119894119895119879119894119895= (

120597

120597119905

+ 1205910

1205972

1205971199052)(119879119900120573119894119895119894119895+ 120588119862119890119879) 119894 119895 = 1 2 3

(2)

The constitutive relations for thermally conducting trans-versely isotropic fiber-reinforced linearly elastic media are

120590119894119895= 120582119890119896119896120575119894119895+ 2120583119879119890119894119895+ 120572 (119886

119896119886119898119890119896119898120575119894119895+ 119886119894119886119895119890119896119896)

+ 2 (120583119871minus 120583119879) (119886119894119886119896119890119896119895+ 119886119895119886119896119890119896119894)

+ 120573 (119886119896119886119898119890119896119898119886119894119886119895) minus 120573119894119895119879120575119894119895

119894 119895 119896 119898 = 1 2 3

(3)

Journal of Thermodynamics 3

where

119890119894119895=

1

2

(119906119894119895+ 119906119895119894) 120596

119894119895=

1

2

(119906119895119894minus 119906119894119895)

119894 119895 = 1 2 3

(4)

and 120582 120583119879are elastic parameters 120572 120573 (120583

119871minus120583119879) are reinforced

anisotropic elastic parameters 120588 is the mass density 120590119894119895

are components of stress tensor 119906119894are the displacement

components 119890119894119895are components of infinitesimal strain 119879

is the temperature change of a material particle 119879119900is the

reference uniform temperature of the body 119896119894119895are coefficients

of thermal conductivity 120573119894119895

are thermal elastic couplingtensor 119862

119890is the specific heat at constant strain 120575

119894119895is the

Kronecker delta and 119875 is the initial Pressure The commanotation is used for spatial derivatives and superimposeddot represents time differentiation 119886

119895are components of a

all referred to Cartesian coordinate The vector a may be afunction of position We choose a so that its components are(1 0 0)

The equation which governs the motion in terms ofvelocity potential 120595 of the homogeneous inviscid liquid isgiven by Ewing et al [31]

1205972120595

1205971199092

1

+

1205972120595

1205971199092

2

=

1

(120572ℓ)2

1205972120595

1205971199052 (5)

where 120572ℓ = radic120582ℓ120588ℓ is the velocity of sound liquid

The displacement components 119906ℓ1 119906ℓ

2and pressure 119901ℓ in

the medium1198722are given by

119906ℓ

1=

120597120595

1205971199091

119906ℓ

2=

120597120595

1205971199092

119901ℓ= minus120588ℓ 120597120595

120597119905

(6)

3 Formulation of the Problem

We consider a homogeneous thermally conducting trans-versely isotropic fiber-reinforced medium (119872

1) with an

initial hydrostatic stress lying under a uniform homogeneousinviscid liquid layer medium (119872

2) of thickness 119867 We take

the rectangular Cartesian coordinate system 119874119909111990921199093at any

point on the plane horizontal surface and 1199091-axis in the

direction of wave propagation and 1199092-axis pointing vertically

downward into the half-space so that all particles on a lineparallel to 119909

3-axis are equally displaced therefore all the field

quantities will be independent of 1199093-coordinates Hence the

liquid layer medium 1198722occupies the region minus119867 lt 119909

2lt 0

and 1199092gt 0 is occupied by the half-space medium 119872

1 The

plane 1199092= minus119867 represents the free surface of the liquid and

1199092= 0 is taken as interface between 119872

1and 119872

2medium

(Figure 1)The displacement components for medium119872

1are taken

as

= (1199061 1199062 0) (7)

Homogeneous inviscid liquid layer

Transversely isotropic fiber-reinforced

0

x2

x2

= minusH

H

x1

(medium M2)

thermal half space (medium M1)

Figure 1 Geometry of the problem

With the help of (3) (4) and (7) and (1)-(2) take the form

11986211

12059721199061

1205971199092

1

+ (11986212+ 1198620minus

119875

2

)

12059721199062

12059711990911205971199092

+ (1198620+

119875

2

)

12059721199061

1205971199092

2

minus 12057311

120597119879

1205971199091

= 120588

12059721199061

1205971199052

(8)

11986222

12059721199062

1205971199092

2

+ (11986212+ 1198620minus

119875

2

)

12059721199061

12059711990911205971199092

+ (1198620+

119875

2

)

12059721199062

1205971199092

1

minus 12057322

120597119879

1205971199092

= 120588

12059721199062

1205971199052

(9)

1205972119879

1205971199092

1

+ 119896

1205972119879

1205971199092

2

minus

120588119862119890

11989611

(

120597119879

120597119905

+ 120591119900

1205972119879

1205971199052)

= 120576 [(

12059721199061

1205971199091120597119905

+ 120591119900

12059731199061

12059711990911205971199052) + 120573(

12059721199062

1205971199092120597119905

+ 120591119900

12059731199062

12059711990921205971199052)]

(10)

where

12057311= (11986211+ 11986212) 12057211+ 1198621212057222

12057322= (11986212+ 11986223minus 11986255) 12057211+ 1198622212057222

11986212= 120582 + 120572 119862

11= 120582 + 2120572 + 4120583

119871minus 2120583119879+ 120573

11986222= 11986233= 120582 + 2120583

119879

11986244= 11986266= 2120583119871 119862

55= 2120583119879

11986223= 11986233minus 11986255 119862

119900=

11986244

2

120576 =

11987911990012057311

11989611

119896 =

11989622

11989611

120573 =

12057322

12057311

(11)

and 120582 120572 120573 120583119871 120583119879are material constants 120572

11 12057222

are com-ponents of linear thermal expansion and 120591

119900is thermal

relaxation time

4 Journal of Thermodynamics

To facilitate the solution the following dimensionlessquantities are introduced

(1199091015840

1 1199091015840

2) =

119908lowast

V1

(1199091 1199092) (119906

1015840

1 1199061015840

2) =

120588V1119908lowast

12057311119879119900

(1199061 1199062)

1199051015840

119894119895=

119905119894119895

12057311119879119900

1198791015840=

119879

119879119900

1199051015840= 119908lowast119905

ℎ1015840=

V1

119908lowastℎ 120591

1015840

119900= 119908lowast120591119900 120595

1015840=

119908lowast120595

V21

(119906ℓ1015840

1 119906ℓ1015840

2) =

119908lowast

V1

(119906ℓ

1 119906ℓ

2) 119901

ℓ1015840

=

119901ℓ

120582ℓ 119905

ℓ1015840

22=

119905ℓ

22

120582ℓ

(12)

where 119908lowast = 1198621198901198621111989611 V21= 11986211120588 and 119908lowast is the character-

istic frequency of the medium

4 Solution of the Problem

We assume the solutions of the form

(1199061 1199062 119879 120595) = (1119882 119877 119878) 119880 exp 119894120585 (119909

1+ 1198981199092minus 119888119905) (13)

where 119888 = 120596120585 is the nondimensional phase velocity 120596 isthe frequency and 120585 is the wave number 119898 is still unknownparameter 1119882 119877 119878 are respectively the amplitude ratios of1199061 1199062 119879 and 119906ℓ

2with respect to 119906

1 Substituting the values of

1199061 1199062and 119879 from (13) in (8) (9) and (10) we obtain

[

119898

11986211

(11986212+ 119862119900minus

119875

2

)]119882 + (

minus1

119894120585

)119877

+ [1 minus 1198882+

1

11986211

(119862119900+

119875

2

)1198982] = 0

[

11986222

11986211

1198982+

1

11986211

(119862119900+

119875

2

) minus 1198882]119882

+ (

minus119898120573

119894120585

)119877 + [

1

11986211

(11986212+ 119862119900minus

119875

2

)119898] = 0

[1205761120573 (minus119898119888 + 119898119888

2120591119900119894120585)]119882 + [1 + 119896119898

2+ (

119888

119894120585

minus 1205911199001198882)]119877

+ 1205761[119888 minus 120591

1199001198882119894120585] = 0

(14)

where

1205761=

1198791199001205732

11

120588119908lowast11989611

(15)

The system of (14) has a nontrivial solution if the deter-minant of the coefficients [1119882 119877]119879 vanishes which yields tothe following polynomial characteristic equation

1198986+ 119889211198984+ 119889311198982+ 11988941= 0 (16)

where

1198891= [119896 (119871

21198921)]

1198892= [minus 119892

11198651198712+ 119896 (119871

11198921+ 11987121198922minus 1198602

1)

+ 11987121198921+ (120573)

2

12057611198651198712]

1198893= [minus 119865119892

1(1198711+ 1205761) minus 119865119892

21198712

+ 1205731198651198601(1 minus 120576

1) + 119896 (119871

11198922) + 11987111198921

+ 11987121198922+ (120573)

2

12057611198651198711+ 1198602

1(119865 minus 1)]

1198894= [minus119892

2119865 (1198711+ 1205761) + 11987111198922]

11988921=

1198892

1198891

11988931=

1198893

1198891

11988941=

1198894

1198891

119865 =

minus119888 + 1205911199001198882119894120585

119894120585

119860 = 1198981198601

1198601=

1

11986211

(11986212+ 119862119900minus

119875

2

)

119866 = 11989211198982+ 1198922 119892

1=

11986222

11986211

1198922=

1

11986211

(119862119900+

119875

2

) minus 1198882

119871 = 1198711+ 11987121198982 119871

1= 1 minus 119888

2

1198712=

1

11986211

(119862119900+

119875

2

)

(17)

The characteristic equation (16) is cubic in1198982 and hencepossesses three roots (1198982

119901 119901 = 1 2 3) Therefore there exist

three types of quasi-waves in transversely isotropic elastichalf-space namely quasi-longitudinal waves (QP) quasi-transverse waves (QSV) and quasi-thermal waves (QT)

The formal expression for displacements and volumefraction field satisfying the radiation condition that Re(119898

119901)ge

0 can be written as

(1199061 1199062 119879) =

3

sum

119901=1

119866119901(1 1198991119901 1198992119901) exp 119894120585 (119909

1+ 1198941198981199011199092minus 119888119905)

(18)

where

1198991119901=

Δ1

Δ

1198992119901=

Δ2

Δ

Δ1= (

minus1205731198981205761119888 (1 minus 120591

119900119888119894120585)

119894120585

minus 119860119884)

Δ2= (119866120576

1119888 (1 minus 120591

119900119888119894120585) minus 119860119885)

Journal of Thermodynamics 5

Δ = (119866119884 +

120573119898119885

119894120585

)

119884 = 1 + 1198961198982+ (

119888

119894120585

minus 1205911199001198882)

119885 = 1205761120573119898119888 (minus1 + 119888120591

119900119894120585)

(19)

and 1198661 1198662 1198663are arbitrary constants

Substituting the values of 120595 from (13) in (5) and aftersimplification we obtain

120595 = [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)] exp (119894120585 (119909

1minus 119888119905))

(20)

where

1198984= plusmn((

V1119888

120572ℓ)

2

minus 1)

12

(21)

and 1198664 1198665are arbitrary constants

The normal displacement and normal stress for medium1198722 with the help of (5) (6) (12) and (20) are given by

119906ℓ

2= 1198984120585 [1198664cos (120585119898

41199092) minus 1198665sin (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

119905ℓ

22= minus 120585

2(1 + 119898

2

4) [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

(22)

Also from (3) we have

11990522= 1198621211990611+ 1198622211990622minus 12057322119879

11990521= 119862119900(11990612+ 11990621)

(23)

5 Boundary Conditions

The appropriate boundary conditions are as follows

(i) vanishing of normal stress component at the freesurface of the liquid layer 119909

2= minus119867 that is

(119905ℓ

22)1198722

= minus119901ℓ= 0 (24)

(ii) continuity of normal stress components at the inter-face 119909

2= 0 that is

(11990522)1198721

= minus119901ℓ (25)

(iii) normal displacement components at the interface1199092= 0 that is

(119906ℓ

2)1198722

= (2)1198721

(26)

(iv) vanishing shear stress for medium1198721at the interface

1199092= 0 that is

(11990521)1198721

= 0 (27)

(v) thermal boundary conditions

120597119879

1205971199092

+ ℎ119879 = 0 (28)

where ℎ 997888rarr 0 corresponds Insulated boundarycondition ℎ 997888rarr infin corresponds Isolated boundarycondition

6 Derivations of the Secular Equations

Substituting the value of 1199062 119879 119906ℓ

2 119905ℓ

22from (18) and (22) in

(24)ndash(28) andwith the aid of (7) and (12) after simplificationwe obtain

1198882= (

120572ℓ

V1

)

2

[

[

(tanminus1 (11987911))

2

+ 12058521198672

12058521198672

]

]

(29)

where

11987911=

119862111198984(119864111198631minus 119864121198632+ 119864131198633)

119894119888120588ℓV21(119899111198631minus 119899121198632+ 119899131198633)

1198641119901= 119894120585 (

11986212

11986211

) + (

11986222

11986211

)1198991119901(1198942120585119898119901) minus (

12057322

12057311

)1198992119901

1198642119901= 120585 (119899

1119901119894 minus 119898119901) 119864

3119901= 1198992119901(ℎ minus 120585119898

119901)

(119901 = 1 2 3)

1198631= 1198642211986433minus 1198643211986423 119863

2= 1198642111986433minus 1198643111986423

1198633= 1198642111986432minus 1198643111986422

(30)

Equation (29) is surface waves secular equation of fiber-reinforced transversely isotropic thermoelastic half-spaceunder a uniform homogeneous inviscid liquid layer Thesecular equation has complete information about the phasevelocity wave number and attenuation coefficient of surfacewaves in such medium

61 Particular Cases

(i) If we take ℎ rarr 0 in (29) we obtain the frequencyequation for the thermally insulated boundary

(ii) If we take ℎ rarr infin in (29) we obtain the frequencyequation for the thermally isolated boundary

(iii) Taking 119875 rarr 0 in secular equation (29) yield thefrequency equation for fiber-reinforced transverselyisotropic thermoelastic half space without initialstress under a uniform homogeneous inviscid liquidlayer

(iv) For 120583119879= 120583119871= 120583 119870

11= 11987022= 119870 120573

11= 12057322= 120573 =

(3120582 + 2120583)120572119905 (29) reduces to the frequency equation

in a homogeneous Isotropic half space under inviscidliquid

(v) 120591119900rarr 0 in (10) we obtained the frequency equation

in fiber-reinforced coupled thermoelastic half spaceunder inviscid liquid layer

6 Journal of Thermodynamics

7 Rayleigh Wave

(1) Taking119867 rarr 0 the frequency equation (29) reduces to

11986411(1198642211986433minus 1198642311986432) minus 11986412(1198642111986433minus 1198643111986423)

+ 11986413(1198642111986432minus 1198643111986422) = 0

(31)

Equation (31) is the frequency equation for Rayleighsurface wave propagation in a fiber-reinforced transverselyisotropic thermoelastic half-space with initial stress In gen-eral wave number and hence phase velocity of waves iscomplex quantity therefore the waves are attenuated in spaceIf we write

119888minus1= Vminus1 + 119894120596minus1119865 (32)

so that 120585 = 119870 + 119894119865 where119870 = 120596V V and 119865 are real Also theroots of characteristic equation (16) are in general complexand hence we assume that119898

119901= 119901119901+119894119902119901 so that the exponent

in the plane wave solutions (18) for the half-space becomes

119894119870 (1199091minus 119898119868

1199011199092minus V119905) minus 119870(

119865

119870

1199091+ 119898119877

1199011199092) (33)

where

119898119877

119901= 119901119901minus 119902119901

119865

119870

119898119868

119901= 119902119901+ 119901119901

119865

119870

(119901 = 1 2 3)

(34)

This shows that V is the propagation velocity and 119865 is theattenuation coefficient of wave

The equation (18) can be rewritten as

1199061=

3

sum

119901=1

119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

1199062=

3

sum

119901=1

1198991119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

119879 =

3

sum

119901=1

1198992119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

(35)

with 120582119901= 119870(119898

119877

119901+ 119894119898119868

119901) = 120582

119877

119901+ 119894120582119868

119901(say) (119901 = 1 2 3)

Moreover it is clear that

10038161003816100381610038161003816120582119877

119901

10038161003816100381610038161003816

2

minus

10038161003816100381610038161003816120582119868

119901

10038161003816100381610038161003816

2

= 1198702[(119898119877

119901)

2

minus (119898119868

119901)

2

] (36)

71 Normal Stress Normal stress at the interface 1199092= 0 is

given by

11990522= (119862

12119894120585

3

sum

119901=1

119866119901minus 11986222120585

3

sum

119901=1

1198981199011198661199011198991119901minus 12057322

3

sum

119901=1

1198661199011198992119901)

times exp (119894120585 (1199091minus V119905)) (119901 = 1 2 3)

(37)

72 Specific Loss The Specific loss is the ratio of energy (Δ119882)dissipated in taking a specimen through a stress cycle to theelastic energy (119882) stored in the specimen when the strainis maximum The Specific loss is the most direct method ofdefining internal friction for amaterial For a sinusoidal planewave of small amplitude Kolsky [32] shows that the specificloss Δ119882119882 equals 4120587 times the absolute value of imaginarypart of 120585 to the real part of 120585 that is

Δ119882

119882

= 4120587

10038161003816100381610038161003816100381610038161003816

Im (120585)

Re (120585)

10038161003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

V119866120596

1003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

119866

119865

1003816100381610038161003816100381610038161003816

(38)

8 Numerical Results and Discussion

For numerical computations we take the following valuesof the relevant parameters for generalized fiber-reinforcedtransversely isotropic thermoelastic solid

120582 = 566 times 1010Nm2 120588 = 2660Kgm3

120583119879= 246 times 10

10Nm2 120583119871= 566 times 10

10Nm2

120572 = minus128 times 1010Nm2 120573 = 22090 times 10

10Nm2

11987011= 00921 times 10

3 Jmminus1 degminus1 sminus1 119875 = 100

11987022= 00963 times 10

3 Jmminus1 degminus1 sminus1

1205721= 0017 times 10

4 degminus1

1205722= 0015 times 10

4 degminus1 119879119900= 293K

120591119900= 005 s ℎ = 1 119867 = 1

119862119890= 0787 times 10

3 JKgminus1degminus1 120596 = 2 sminus1(39)

The physical constants for water are given by Ewing et al[31]

120588ℓ= 10 times 10

3 Kgm3 120582ℓ= 214 times 10

3Nm2 (40)

For comparison with the generalized fiber-reinforcedisotropic thermoelastic solid following Ezzat [33] we take

Journal of Thermodynamics 7

0 1 2 3 4 5

Phas

e velo

city

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

25

20

15

10

05

00

times101

Figure 2 Variation of phase velocity wrt wave number (with andwithout initial stress)

the following values of relevant parameters for the case ofcopper material as

120582 = 776 times 1010Nm2

120583119879= 120583119871= 120583 = 386 times 10

10Nm2

11987011= 11987022= 119870 = 386NKminus1 sminus1

12057311= 12057322= 120573 = (3120582 + 2120583) 120572

119905

120572119905= 178 times 10

minus5 Kminus1

1205721= 1005 degminus1 120572

2= 1001 degminus1

119862119890= 3831m2 Kminus1 sminus1 119875 = 0

119867 = 1 120588 = 8954Kgm3

120591119900= 01 s ℎ = 1

(41)

Using the above values of parameters the dispersioncurves of nondimensional phase velocity attenuation coef-ficient and specific heat are shown graphically with respectto nondimensional wave number Here we consider abbre-viations FTTIWIS FTTIIS FTIWIS and FTIIS for fiber-reinforced transversely isotropic thermoelasticmaterial with-out initial stress fiber-reinforced transversely isotropic ther-moelastic material with initial stress fiber-reinforced iso-tropic thermoelastic material without initial stress and fiber-reinforced isotropic thermoelastic material with initial stressrespectively and119867 refers to the thickness of the layer

Figure 2 depicts the variations of phase velocity withrespect to the wave number The behavior of phase velocity

0 1 2 3 4 5Wave number

05

00

minus05

minus10

minus15

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times10minus1

Figure 3 Variation of attenuation coefficient wrt wave number(with and without initial stress)

for FTTIWIS FTTIIS FTIWIS and FTIIS is similar butthe corresponding values are different in magnitude Thevalues of phase velocity decrease for lower wave numberand become stable for higher wave number in all the casesFigure 3 shows the variation of attenuation coefficient withrespect to the wave number For FTTIWIS and FTTIIS thevalues of attenuation coefficient increase monotonically upto wave number (120585) = 2 and after that it becomes constant forwhole range of wave number On the other hand the values ofattenuation coefficients are consistent within the whole rangeof wave number for FTIWIS and FTIIS

Figure 4 indicates the variation of phase velocity withrespect to thewave number for different values of thickness oflayer Here we consider the particular case of fiber-reinforcedtransversely isotropic thermoelastic material and analyses ofthe behavior of curves with respect to initial stress It isobserved that for small value of wave number the graph ofphase velocity shows opposite behavior wrt thickness thatis on increasing the values of thickness of layer the values ofcorresponding phase velocity decreases and it became steadyfor large values of wave number

Figure 5 shows the variation of attenuation coefficientwith respect to the wave number for different value ofthickness of the layer Here also we considered the particularcase of fiber-reinforced transversely isotropic thermoelasticmaterial and analyses of the behavior of curveswith respect toinitial stress It is perceived that on decreasing the value of119867the curve shows consistency for small value of wave numberand on increasing the value of119867 the curve shows uniformityfor large value of 120585

Figure 6 depicts the variations of specific loss verseswave number For FTIWIS and FTIIS the curve shows sharpdecrease in the value of specific loss for the range 0 le 120585 le 1and become parallel for 120585 gt 1 It is observed that graphs

8 Journal of Thermodynamics

0 1 2 3 4 5Wave number

25

20

15

10

05

00

Phas

e velo

city

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

times102

Figure 4 Variation of phase velocity wrt wave number (with andwithout initial stress for different value of119867)

corresponding to FTTIWIS and FTTIIS follow the similarpattern for whole range of 120585 but the values are different inmagnitude

81 Rayleigh Wave Figure 7 shows the variations of phasevelocity of Rayleigh wave verses wave number for FTTIWISFTTIIS FTIWIS and FTIIS The value of phase velocitydecreases for small values of 120585 and become stable for highervalue of 120585 It is analyzed that in all the cases behavior of curvesare almost the same within the range 2 le 120585 le 5 Figure 8shows the variations of attenuation coefficient of Rayleighwave verses wave number for FTTIWIS FTTIIS FTIWISand FTIIS For FTIWIS and FTIIS the graph shows a suddenfall for 120585 = 05 and then increases monotonically up to 120585 = 2and becomes constant for the remaining range of 120585 A largedifference between the behavior of curves for the transverselyisotropic and isotropic case is also perceived

9 Conclusions

The propagation of surface waves in an initially stressedfiber-reinforced transversely isotropic thermoelastic mate-rial under an inviscid liquid layer has been studied Afterdeveloping the formal solutions the secular equation forsurface wave propagation is derived Some special casesof frequency equation are also discussed The attenuationcoefficients and specific loss (liquid layer and half-space)have also been studied and computed for FTTIWIS FTTIISFTIWIS and FTIISThe trend of phase velocity is the same inall these four cases for surface wave but varies for attenuation

0 1 2 3 4 5Wave number

00

05

times10minus1

minus15

minus05

minus10

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

Figure 5 Variation of attenuation coeff wrt wave number (withand without initial stress for different value of119867)

0 1 2 3 4 5

Spec

ific l

oss

40

35

30

25

20

15

10

05

00

minus05

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

times10minus4

Figure 6 Variation of specific loss wrt wave number (with andwithout initial stress)

coefficient From numerical analysis it is shown that thephase speeds and attenuation coefficients for Rayleigh waveare also affected significantly due to the presence of initialstress parameters

Journal of Thermodynamics 9

0 1 2 3 4 5

Phas

e velo

city

Wave number

12

10

80

times103

60

40

20

00

FTTIWISFTTIIS

FTIWISFTIIS

Figure 7 Variation of phase velocity wrt wave number forRayleigh wave (with and without initial stress)

0 1 2 3 4 5

00

minus20

minus40

minus60

minus80

minus10

Wave number

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times105

Figure 8 Variation of attenuation coeff wrt wave number forRayleigh wave (with and without initial stress)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A J M Spencer Deformation of Fibre-reinforced MaterialsUniversity of Oxford Clarendon Va USA 1941

[2] A J Belfield T G Rogers and A JM Spencer ldquoStress in elasticplates reinforced by fibres lying in concentric circlesrdquo Journal oftheMechanics and Physics of Solids vol 31 no 1 pp 25ndash54 1983

[3] A C Pipkin ldquoFinite deformations of ideal fiber-reinforcedcompositesrdquo in Composites Materials G P Sendeckyj Ed vol2 of Mechanics of Composite Materials pp 251ndash308 AcademicPress New York NY USA 1973

[4] P R Sengupta and S Nath ldquoSurface waves in fibre-reinforcedanisotropic elastic mediardquo Indian Academy of Sciences SadhanaProceedings vol 26 pp 363ndash370 2001

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

[6] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 198081

[7] A U Erdem ldquoHeat Conduction in fiber-reinforced rigid bod-iesrdquo 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK) 6ndash8Eylul Ankara Turkey 1995

[8] R Kumar and R R Gupta ldquoStudy of wave motion in ananisotropic fiber-reinforced thermoelastic solidrdquo Journal ofSolid Mechanics vol 2 no 1 pp 91ndash100 2010

[9] P Chadwick and L T C Seet ldquoWave propagation in atransversely isotropic heat-conducting elastic materialrdquo Math-ematika vol 17 pp 255ndash274 1970

[10] H Singh and J N Sharma ldquoGeneralized thermoelastic waves intransversely isotropicmediardquo Journal of the Acoustical Society ofAmerica vol 77 pp 1046ndash1053 1985

[11] B Singh ldquoWave propagation in an anisotropic generalizedthermoelastic solidrdquo Indian Journal of Pure and Applied Mathe-matics vol 34 no 10 pp 1479ndash1485 2003

[12] P Chadwick Progress in Solid Mechanics vol 1 North-HollandAmsterdam The Netherlands 1960 edited by R Hill I NSneddon

[13] P Chadwick and DWWindle ldquoPropagation of Rayleigh wavesalong isothermal and insulated boundariesrdquo Proceedings of theRoyal Society of America vol 280 pp 47ndash71 1964

[14] J N Sharma and H Singh ldquoThermoelastic surface waves ina transversely isotropic half space with thermal relaxationsrdquoIndian Journal of Pure and Applied Mathematics vol 16 no 10pp 1202ndash1219 1985

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

[16] J Wang and P Slattery ldquoThermoelasticity without energydissipation for initially stressed bodiesrdquo International Journal ofMathematics and Mathematical Sciences vol 31 no 6 pp 329ndash337 2002

[17] D Iesan ldquoA theory of prestressed thermoelastic Cosserat con-tinuardquo Journal of Applied Mathematics and Mechanics vol 88no 4 pp 306ndash319 2008

[18] K Ames and B Straughan ldquoContinuous dependence results forinitially prestressed thermoelastic bodiesrdquo International Journalof Engineering Science vol 30 no 1 pp 7ndash13 1992

[19] J Wang R S Dhaliwal and S R Majumdar ldquoSome theoremsin the generalized theory of thermoelasticity for prestressedbodiesrdquo Indian Journal of Pure and Applied Mathematics vol28 no 2 pp 267ndash276 1997

[20] M Marin and C Marinescu ldquoThermoelasticity of initiallystressed bodies asymptotic equipartition of energiesrdquo Interna-tional Journal of Engineering Science vol 36 no 1 pp 73ndash861998

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

Journal of Thermodynamics 3

where

119890119894119895=

1

2

(119906119894119895+ 119906119895119894) 120596

119894119895=

1

2

(119906119895119894minus 119906119894119895)

119894 119895 = 1 2 3

(4)

and 120582 120583119879are elastic parameters 120572 120573 (120583

119871minus120583119879) are reinforced

anisotropic elastic parameters 120588 is the mass density 120590119894119895

are components of stress tensor 119906119894are the displacement

components 119890119894119895are components of infinitesimal strain 119879

is the temperature change of a material particle 119879119900is the

reference uniform temperature of the body 119896119894119895are coefficients

of thermal conductivity 120573119894119895

are thermal elastic couplingtensor 119862

119890is the specific heat at constant strain 120575

119894119895is the

Kronecker delta and 119875 is the initial Pressure The commanotation is used for spatial derivatives and superimposeddot represents time differentiation 119886

119895are components of a

all referred to Cartesian coordinate The vector a may be afunction of position We choose a so that its components are(1 0 0)

The equation which governs the motion in terms ofvelocity potential 120595 of the homogeneous inviscid liquid isgiven by Ewing et al [31]

1205972120595

1205971199092

1

+

1205972120595

1205971199092

2

=

1

(120572ℓ)2

1205972120595

1205971199052 (5)

where 120572ℓ = radic120582ℓ120588ℓ is the velocity of sound liquid

The displacement components 119906ℓ1 119906ℓ

2and pressure 119901ℓ in

the medium1198722are given by

119906ℓ

1=

120597120595

1205971199091

119906ℓ

2=

120597120595

1205971199092

119901ℓ= minus120588ℓ 120597120595

120597119905

(6)

3 Formulation of the Problem

We consider a homogeneous thermally conducting trans-versely isotropic fiber-reinforced medium (119872

1) with an

initial hydrostatic stress lying under a uniform homogeneousinviscid liquid layer medium (119872

2) of thickness 119867 We take

the rectangular Cartesian coordinate system 119874119909111990921199093at any

point on the plane horizontal surface and 1199091-axis in the

direction of wave propagation and 1199092-axis pointing vertically

downward into the half-space so that all particles on a lineparallel to 119909

3-axis are equally displaced therefore all the field

quantities will be independent of 1199093-coordinates Hence the

liquid layer medium 1198722occupies the region minus119867 lt 119909

2lt 0

and 1199092gt 0 is occupied by the half-space medium 119872

1 The

plane 1199092= minus119867 represents the free surface of the liquid and

1199092= 0 is taken as interface between 119872

1and 119872

2medium

(Figure 1)The displacement components for medium119872

1are taken

as

= (1199061 1199062 0) (7)

Homogeneous inviscid liquid layer

Transversely isotropic fiber-reinforced

0

x2

x2

= minusH

H

x1

(medium M2)

thermal half space (medium M1)

Figure 1 Geometry of the problem

With the help of (3) (4) and (7) and (1)-(2) take the form

11986211

12059721199061

1205971199092

1

+ (11986212+ 1198620minus

119875

2

)

12059721199062

12059711990911205971199092

+ (1198620+

119875

2

)

12059721199061

1205971199092

2

minus 12057311

120597119879

1205971199091

= 120588

12059721199061

1205971199052

(8)

11986222

12059721199062

1205971199092

2

+ (11986212+ 1198620minus

119875

2

)

12059721199061

12059711990911205971199092

+ (1198620+

119875

2

)

12059721199062

1205971199092

1

minus 12057322

120597119879

1205971199092

= 120588

12059721199062

1205971199052

(9)

1205972119879

1205971199092

1

+ 119896

1205972119879

1205971199092

2

minus

120588119862119890

11989611

(

120597119879

120597119905

+ 120591119900

1205972119879

1205971199052)

= 120576 [(

12059721199061

1205971199091120597119905

+ 120591119900

12059731199061

12059711990911205971199052) + 120573(

12059721199062

1205971199092120597119905

+ 120591119900

12059731199062

12059711990921205971199052)]

(10)

where

12057311= (11986211+ 11986212) 12057211+ 1198621212057222

12057322= (11986212+ 11986223minus 11986255) 12057211+ 1198622212057222

11986212= 120582 + 120572 119862

11= 120582 + 2120572 + 4120583

119871minus 2120583119879+ 120573

11986222= 11986233= 120582 + 2120583

119879

11986244= 11986266= 2120583119871 119862

55= 2120583119879

11986223= 11986233minus 11986255 119862

119900=

11986244

2

120576 =

11987911990012057311

11989611

119896 =

11989622

11989611

120573 =

12057322

12057311

(11)

and 120582 120572 120573 120583119871 120583119879are material constants 120572

11 12057222

are com-ponents of linear thermal expansion and 120591

119900is thermal

relaxation time

4 Journal of Thermodynamics

To facilitate the solution the following dimensionlessquantities are introduced

(1199091015840

1 1199091015840

2) =

119908lowast

V1

(1199091 1199092) (119906

1015840

1 1199061015840

2) =

120588V1119908lowast

12057311119879119900

(1199061 1199062)

1199051015840

119894119895=

119905119894119895

12057311119879119900

1198791015840=

119879

119879119900

1199051015840= 119908lowast119905

ℎ1015840=

V1

119908lowastℎ 120591

1015840

119900= 119908lowast120591119900 120595

1015840=

119908lowast120595

V21

(119906ℓ1015840

1 119906ℓ1015840

2) =

119908lowast

V1

(119906ℓ

1 119906ℓ

2) 119901

ℓ1015840

=

119901ℓ

120582ℓ 119905

ℓ1015840

22=

119905ℓ

22

120582ℓ

(12)

where 119908lowast = 1198621198901198621111989611 V21= 11986211120588 and 119908lowast is the character-

istic frequency of the medium

4 Solution of the Problem

We assume the solutions of the form

(1199061 1199062 119879 120595) = (1119882 119877 119878) 119880 exp 119894120585 (119909

1+ 1198981199092minus 119888119905) (13)

where 119888 = 120596120585 is the nondimensional phase velocity 120596 isthe frequency and 120585 is the wave number 119898 is still unknownparameter 1119882 119877 119878 are respectively the amplitude ratios of1199061 1199062 119879 and 119906ℓ

2with respect to 119906

1 Substituting the values of

1199061 1199062and 119879 from (13) in (8) (9) and (10) we obtain

[

119898

11986211

(11986212+ 119862119900minus

119875

2

)]119882 + (

minus1

119894120585

)119877

+ [1 minus 1198882+

1

11986211

(119862119900+

119875

2

)1198982] = 0

[

11986222

11986211

1198982+

1

11986211

(119862119900+

119875

2

) minus 1198882]119882

+ (

minus119898120573

119894120585

)119877 + [

1

11986211

(11986212+ 119862119900minus

119875

2

)119898] = 0

[1205761120573 (minus119898119888 + 119898119888

2120591119900119894120585)]119882 + [1 + 119896119898

2+ (

119888

119894120585

minus 1205911199001198882)]119877

+ 1205761[119888 minus 120591

1199001198882119894120585] = 0

(14)

where

1205761=

1198791199001205732

11

120588119908lowast11989611

(15)

The system of (14) has a nontrivial solution if the deter-minant of the coefficients [1119882 119877]119879 vanishes which yields tothe following polynomial characteristic equation

1198986+ 119889211198984+ 119889311198982+ 11988941= 0 (16)

where

1198891= [119896 (119871

21198921)]

1198892= [minus 119892

11198651198712+ 119896 (119871

11198921+ 11987121198922minus 1198602

1)

+ 11987121198921+ (120573)

2

12057611198651198712]

1198893= [minus 119865119892

1(1198711+ 1205761) minus 119865119892

21198712

+ 1205731198651198601(1 minus 120576

1) + 119896 (119871

11198922) + 11987111198921

+ 11987121198922+ (120573)

2

12057611198651198711+ 1198602

1(119865 minus 1)]

1198894= [minus119892

2119865 (1198711+ 1205761) + 11987111198922]

11988921=

1198892

1198891

11988931=

1198893

1198891

11988941=

1198894

1198891

119865 =

minus119888 + 1205911199001198882119894120585

119894120585

119860 = 1198981198601

1198601=

1

11986211

(11986212+ 119862119900minus

119875

2

)

119866 = 11989211198982+ 1198922 119892

1=

11986222

11986211

1198922=

1

11986211

(119862119900+

119875

2

) minus 1198882

119871 = 1198711+ 11987121198982 119871

1= 1 minus 119888

2

1198712=

1

11986211

(119862119900+

119875

2

)

(17)

The characteristic equation (16) is cubic in1198982 and hencepossesses three roots (1198982

119901 119901 = 1 2 3) Therefore there exist

three types of quasi-waves in transversely isotropic elastichalf-space namely quasi-longitudinal waves (QP) quasi-transverse waves (QSV) and quasi-thermal waves (QT)

The formal expression for displacements and volumefraction field satisfying the radiation condition that Re(119898

119901)ge

0 can be written as

(1199061 1199062 119879) =

3

sum

119901=1

119866119901(1 1198991119901 1198992119901) exp 119894120585 (119909

1+ 1198941198981199011199092minus 119888119905)

(18)

where

1198991119901=

Δ1

Δ

1198992119901=

Δ2

Δ

Δ1= (

minus1205731198981205761119888 (1 minus 120591

119900119888119894120585)

119894120585

minus 119860119884)

Δ2= (119866120576

1119888 (1 minus 120591

119900119888119894120585) minus 119860119885)

Journal of Thermodynamics 5

Δ = (119866119884 +

120573119898119885

119894120585

)

119884 = 1 + 1198961198982+ (

119888

119894120585

minus 1205911199001198882)

119885 = 1205761120573119898119888 (minus1 + 119888120591

119900119894120585)

(19)

and 1198661 1198662 1198663are arbitrary constants

Substituting the values of 120595 from (13) in (5) and aftersimplification we obtain

120595 = [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)] exp (119894120585 (119909

1minus 119888119905))

(20)

where

1198984= plusmn((

V1119888

120572ℓ)

2

minus 1)

12

(21)

and 1198664 1198665are arbitrary constants

The normal displacement and normal stress for medium1198722 with the help of (5) (6) (12) and (20) are given by

119906ℓ

2= 1198984120585 [1198664cos (120585119898

41199092) minus 1198665sin (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

119905ℓ

22= minus 120585

2(1 + 119898

2

4) [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

(22)

Also from (3) we have

11990522= 1198621211990611+ 1198622211990622minus 12057322119879

11990521= 119862119900(11990612+ 11990621)

(23)

5 Boundary Conditions

The appropriate boundary conditions are as follows

(i) vanishing of normal stress component at the freesurface of the liquid layer 119909

2= minus119867 that is

(119905ℓ

22)1198722

= minus119901ℓ= 0 (24)

(ii) continuity of normal stress components at the inter-face 119909

2= 0 that is

(11990522)1198721

= minus119901ℓ (25)

(iii) normal displacement components at the interface1199092= 0 that is

(119906ℓ

2)1198722

= (2)1198721

(26)

(iv) vanishing shear stress for medium1198721at the interface

1199092= 0 that is

(11990521)1198721

= 0 (27)

(v) thermal boundary conditions

120597119879

1205971199092

+ ℎ119879 = 0 (28)

where ℎ 997888rarr 0 corresponds Insulated boundarycondition ℎ 997888rarr infin corresponds Isolated boundarycondition

6 Derivations of the Secular Equations

Substituting the value of 1199062 119879 119906ℓ

2 119905ℓ

22from (18) and (22) in

(24)ndash(28) andwith the aid of (7) and (12) after simplificationwe obtain

1198882= (

120572ℓ

V1

)

2

[

[

(tanminus1 (11987911))

2

+ 12058521198672

12058521198672

]

]

(29)

where

11987911=

119862111198984(119864111198631minus 119864121198632+ 119864131198633)

119894119888120588ℓV21(119899111198631minus 119899121198632+ 119899131198633)

1198641119901= 119894120585 (

11986212

11986211

) + (

11986222

11986211

)1198991119901(1198942120585119898119901) minus (

12057322

12057311

)1198992119901

1198642119901= 120585 (119899

1119901119894 minus 119898119901) 119864

3119901= 1198992119901(ℎ minus 120585119898

119901)

(119901 = 1 2 3)

1198631= 1198642211986433minus 1198643211986423 119863

2= 1198642111986433minus 1198643111986423

1198633= 1198642111986432minus 1198643111986422

(30)

Equation (29) is surface waves secular equation of fiber-reinforced transversely isotropic thermoelastic half-spaceunder a uniform homogeneous inviscid liquid layer Thesecular equation has complete information about the phasevelocity wave number and attenuation coefficient of surfacewaves in such medium

61 Particular Cases

(i) If we take ℎ rarr 0 in (29) we obtain the frequencyequation for the thermally insulated boundary

(ii) If we take ℎ rarr infin in (29) we obtain the frequencyequation for the thermally isolated boundary

(iii) Taking 119875 rarr 0 in secular equation (29) yield thefrequency equation for fiber-reinforced transverselyisotropic thermoelastic half space without initialstress under a uniform homogeneous inviscid liquidlayer

(iv) For 120583119879= 120583119871= 120583 119870

11= 11987022= 119870 120573

11= 12057322= 120573 =

(3120582 + 2120583)120572119905 (29) reduces to the frequency equation

in a homogeneous Isotropic half space under inviscidliquid

(v) 120591119900rarr 0 in (10) we obtained the frequency equation

in fiber-reinforced coupled thermoelastic half spaceunder inviscid liquid layer

6 Journal of Thermodynamics

7 Rayleigh Wave

(1) Taking119867 rarr 0 the frequency equation (29) reduces to

11986411(1198642211986433minus 1198642311986432) minus 11986412(1198642111986433minus 1198643111986423)

+ 11986413(1198642111986432minus 1198643111986422) = 0

(31)

Equation (31) is the frequency equation for Rayleighsurface wave propagation in a fiber-reinforced transverselyisotropic thermoelastic half-space with initial stress In gen-eral wave number and hence phase velocity of waves iscomplex quantity therefore the waves are attenuated in spaceIf we write

119888minus1= Vminus1 + 119894120596minus1119865 (32)

so that 120585 = 119870 + 119894119865 where119870 = 120596V V and 119865 are real Also theroots of characteristic equation (16) are in general complexand hence we assume that119898

119901= 119901119901+119894119902119901 so that the exponent

in the plane wave solutions (18) for the half-space becomes

119894119870 (1199091minus 119898119868

1199011199092minus V119905) minus 119870(

119865

119870

1199091+ 119898119877

1199011199092) (33)

where

119898119877

119901= 119901119901minus 119902119901

119865

119870

119898119868

119901= 119902119901+ 119901119901

119865

119870

(119901 = 1 2 3)

(34)

This shows that V is the propagation velocity and 119865 is theattenuation coefficient of wave

The equation (18) can be rewritten as

1199061=

3

sum

119901=1

119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

1199062=

3

sum

119901=1

1198991119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

119879 =

3

sum

119901=1

1198992119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

(35)

with 120582119901= 119870(119898

119877

119901+ 119894119898119868

119901) = 120582

119877

119901+ 119894120582119868

119901(say) (119901 = 1 2 3)

Moreover it is clear that

10038161003816100381610038161003816120582119877

119901

10038161003816100381610038161003816

2

minus

10038161003816100381610038161003816120582119868

119901

10038161003816100381610038161003816

2

= 1198702[(119898119877

119901)

2

minus (119898119868

119901)

2

] (36)

71 Normal Stress Normal stress at the interface 1199092= 0 is

given by

11990522= (119862

12119894120585

3

sum

119901=1

119866119901minus 11986222120585

3

sum

119901=1

1198981199011198661199011198991119901minus 12057322

3

sum

119901=1

1198661199011198992119901)

times exp (119894120585 (1199091minus V119905)) (119901 = 1 2 3)

(37)

72 Specific Loss The Specific loss is the ratio of energy (Δ119882)dissipated in taking a specimen through a stress cycle to theelastic energy (119882) stored in the specimen when the strainis maximum The Specific loss is the most direct method ofdefining internal friction for amaterial For a sinusoidal planewave of small amplitude Kolsky [32] shows that the specificloss Δ119882119882 equals 4120587 times the absolute value of imaginarypart of 120585 to the real part of 120585 that is

Δ119882

119882

= 4120587

10038161003816100381610038161003816100381610038161003816

Im (120585)

Re (120585)

10038161003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

V119866120596

1003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

119866

119865

1003816100381610038161003816100381610038161003816

(38)

8 Numerical Results and Discussion

For numerical computations we take the following valuesof the relevant parameters for generalized fiber-reinforcedtransversely isotropic thermoelastic solid

120582 = 566 times 1010Nm2 120588 = 2660Kgm3

120583119879= 246 times 10

10Nm2 120583119871= 566 times 10

10Nm2

120572 = minus128 times 1010Nm2 120573 = 22090 times 10

10Nm2

11987011= 00921 times 10

3 Jmminus1 degminus1 sminus1 119875 = 100

11987022= 00963 times 10

3 Jmminus1 degminus1 sminus1

1205721= 0017 times 10

4 degminus1

1205722= 0015 times 10

4 degminus1 119879119900= 293K

120591119900= 005 s ℎ = 1 119867 = 1

119862119890= 0787 times 10

3 JKgminus1degminus1 120596 = 2 sminus1(39)

The physical constants for water are given by Ewing et al[31]

120588ℓ= 10 times 10

3 Kgm3 120582ℓ= 214 times 10

3Nm2 (40)

For comparison with the generalized fiber-reinforcedisotropic thermoelastic solid following Ezzat [33] we take

Journal of Thermodynamics 7

0 1 2 3 4 5

Phas

e velo

city

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

25

20

15

10

05

00

times101

Figure 2 Variation of phase velocity wrt wave number (with andwithout initial stress)

the following values of relevant parameters for the case ofcopper material as

120582 = 776 times 1010Nm2

120583119879= 120583119871= 120583 = 386 times 10

10Nm2

11987011= 11987022= 119870 = 386NKminus1 sminus1

12057311= 12057322= 120573 = (3120582 + 2120583) 120572

119905

120572119905= 178 times 10

minus5 Kminus1

1205721= 1005 degminus1 120572

2= 1001 degminus1

119862119890= 3831m2 Kminus1 sminus1 119875 = 0

119867 = 1 120588 = 8954Kgm3

120591119900= 01 s ℎ = 1

(41)

Using the above values of parameters the dispersioncurves of nondimensional phase velocity attenuation coef-ficient and specific heat are shown graphically with respectto nondimensional wave number Here we consider abbre-viations FTTIWIS FTTIIS FTIWIS and FTIIS for fiber-reinforced transversely isotropic thermoelasticmaterial with-out initial stress fiber-reinforced transversely isotropic ther-moelastic material with initial stress fiber-reinforced iso-tropic thermoelastic material without initial stress and fiber-reinforced isotropic thermoelastic material with initial stressrespectively and119867 refers to the thickness of the layer

Figure 2 depicts the variations of phase velocity withrespect to the wave number The behavior of phase velocity

0 1 2 3 4 5Wave number

05

00

minus05

minus10

minus15

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times10minus1

Figure 3 Variation of attenuation coefficient wrt wave number(with and without initial stress)

for FTTIWIS FTTIIS FTIWIS and FTIIS is similar butthe corresponding values are different in magnitude Thevalues of phase velocity decrease for lower wave numberand become stable for higher wave number in all the casesFigure 3 shows the variation of attenuation coefficient withrespect to the wave number For FTTIWIS and FTTIIS thevalues of attenuation coefficient increase monotonically upto wave number (120585) = 2 and after that it becomes constant forwhole range of wave number On the other hand the values ofattenuation coefficients are consistent within the whole rangeof wave number for FTIWIS and FTIIS

Figure 4 indicates the variation of phase velocity withrespect to thewave number for different values of thickness oflayer Here we consider the particular case of fiber-reinforcedtransversely isotropic thermoelastic material and analyses ofthe behavior of curves with respect to initial stress It isobserved that for small value of wave number the graph ofphase velocity shows opposite behavior wrt thickness thatis on increasing the values of thickness of layer the values ofcorresponding phase velocity decreases and it became steadyfor large values of wave number

Figure 5 shows the variation of attenuation coefficientwith respect to the wave number for different value ofthickness of the layer Here also we considered the particularcase of fiber-reinforced transversely isotropic thermoelasticmaterial and analyses of the behavior of curveswith respect toinitial stress It is perceived that on decreasing the value of119867the curve shows consistency for small value of wave numberand on increasing the value of119867 the curve shows uniformityfor large value of 120585

Figure 6 depicts the variations of specific loss verseswave number For FTIWIS and FTIIS the curve shows sharpdecrease in the value of specific loss for the range 0 le 120585 le 1and become parallel for 120585 gt 1 It is observed that graphs

8 Journal of Thermodynamics

0 1 2 3 4 5Wave number

25

20

15

10

05

00

Phas

e velo

city

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

times102

Figure 4 Variation of phase velocity wrt wave number (with andwithout initial stress for different value of119867)

corresponding to FTTIWIS and FTTIIS follow the similarpattern for whole range of 120585 but the values are different inmagnitude

81 Rayleigh Wave Figure 7 shows the variations of phasevelocity of Rayleigh wave verses wave number for FTTIWISFTTIIS FTIWIS and FTIIS The value of phase velocitydecreases for small values of 120585 and become stable for highervalue of 120585 It is analyzed that in all the cases behavior of curvesare almost the same within the range 2 le 120585 le 5 Figure 8shows the variations of attenuation coefficient of Rayleighwave verses wave number for FTTIWIS FTTIIS FTIWISand FTIIS For FTIWIS and FTIIS the graph shows a suddenfall for 120585 = 05 and then increases monotonically up to 120585 = 2and becomes constant for the remaining range of 120585 A largedifference between the behavior of curves for the transverselyisotropic and isotropic case is also perceived

9 Conclusions

The propagation of surface waves in an initially stressedfiber-reinforced transversely isotropic thermoelastic mate-rial under an inviscid liquid layer has been studied Afterdeveloping the formal solutions the secular equation forsurface wave propagation is derived Some special casesof frequency equation are also discussed The attenuationcoefficients and specific loss (liquid layer and half-space)have also been studied and computed for FTTIWIS FTTIISFTIWIS and FTIISThe trend of phase velocity is the same inall these four cases for surface wave but varies for attenuation

0 1 2 3 4 5Wave number

00

05

times10minus1

minus15

minus05

minus10

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

Figure 5 Variation of attenuation coeff wrt wave number (withand without initial stress for different value of119867)

0 1 2 3 4 5

Spec

ific l

oss

40

35

30

25

20

15

10

05

00

minus05

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

times10minus4

Figure 6 Variation of specific loss wrt wave number (with andwithout initial stress)

coefficient From numerical analysis it is shown that thephase speeds and attenuation coefficients for Rayleigh waveare also affected significantly due to the presence of initialstress parameters

Journal of Thermodynamics 9

0 1 2 3 4 5

Phas

e velo

city

Wave number

12

10

80

times103

60

40

20

00

FTTIWISFTTIIS

FTIWISFTIIS

Figure 7 Variation of phase velocity wrt wave number forRayleigh wave (with and without initial stress)

0 1 2 3 4 5

00

minus20

minus40

minus60

minus80

minus10

Wave number

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times105

Figure 8 Variation of attenuation coeff wrt wave number forRayleigh wave (with and without initial stress)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A J M Spencer Deformation of Fibre-reinforced MaterialsUniversity of Oxford Clarendon Va USA 1941

[2] A J Belfield T G Rogers and A JM Spencer ldquoStress in elasticplates reinforced by fibres lying in concentric circlesrdquo Journal oftheMechanics and Physics of Solids vol 31 no 1 pp 25ndash54 1983

[3] A C Pipkin ldquoFinite deformations of ideal fiber-reinforcedcompositesrdquo in Composites Materials G P Sendeckyj Ed vol2 of Mechanics of Composite Materials pp 251ndash308 AcademicPress New York NY USA 1973

[4] P R Sengupta and S Nath ldquoSurface waves in fibre-reinforcedanisotropic elastic mediardquo Indian Academy of Sciences SadhanaProceedings vol 26 pp 363ndash370 2001

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

[6] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 198081

[7] A U Erdem ldquoHeat Conduction in fiber-reinforced rigid bod-iesrdquo 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK) 6ndash8Eylul Ankara Turkey 1995

[8] R Kumar and R R Gupta ldquoStudy of wave motion in ananisotropic fiber-reinforced thermoelastic solidrdquo Journal ofSolid Mechanics vol 2 no 1 pp 91ndash100 2010

[9] P Chadwick and L T C Seet ldquoWave propagation in atransversely isotropic heat-conducting elastic materialrdquo Math-ematika vol 17 pp 255ndash274 1970

[10] H Singh and J N Sharma ldquoGeneralized thermoelastic waves intransversely isotropicmediardquo Journal of the Acoustical Society ofAmerica vol 77 pp 1046ndash1053 1985

[11] B Singh ldquoWave propagation in an anisotropic generalizedthermoelastic solidrdquo Indian Journal of Pure and Applied Mathe-matics vol 34 no 10 pp 1479ndash1485 2003

[12] P Chadwick Progress in Solid Mechanics vol 1 North-HollandAmsterdam The Netherlands 1960 edited by R Hill I NSneddon

[13] P Chadwick and DWWindle ldquoPropagation of Rayleigh wavesalong isothermal and insulated boundariesrdquo Proceedings of theRoyal Society of America vol 280 pp 47ndash71 1964

[14] J N Sharma and H Singh ldquoThermoelastic surface waves ina transversely isotropic half space with thermal relaxationsrdquoIndian Journal of Pure and Applied Mathematics vol 16 no 10pp 1202ndash1219 1985

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

[16] J Wang and P Slattery ldquoThermoelasticity without energydissipation for initially stressed bodiesrdquo International Journal ofMathematics and Mathematical Sciences vol 31 no 6 pp 329ndash337 2002

[17] D Iesan ldquoA theory of prestressed thermoelastic Cosserat con-tinuardquo Journal of Applied Mathematics and Mechanics vol 88no 4 pp 306ndash319 2008

[18] K Ames and B Straughan ldquoContinuous dependence results forinitially prestressed thermoelastic bodiesrdquo International Journalof Engineering Science vol 30 no 1 pp 7ndash13 1992

[19] J Wang R S Dhaliwal and S R Majumdar ldquoSome theoremsin the generalized theory of thermoelasticity for prestressedbodiesrdquo Indian Journal of Pure and Applied Mathematics vol28 no 2 pp 267ndash276 1997

[20] M Marin and C Marinescu ldquoThermoelasticity of initiallystressed bodies asymptotic equipartition of energiesrdquo Interna-tional Journal of Engineering Science vol 36 no 1 pp 73ndash861998

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

4 Journal of Thermodynamics

To facilitate the solution the following dimensionlessquantities are introduced

(1199091015840

1 1199091015840

2) =

119908lowast

V1

(1199091 1199092) (119906

1015840

1 1199061015840

2) =

120588V1119908lowast

12057311119879119900

(1199061 1199062)

1199051015840

119894119895=

119905119894119895

12057311119879119900

1198791015840=

119879

119879119900

1199051015840= 119908lowast119905

ℎ1015840=

V1

119908lowastℎ 120591

1015840

119900= 119908lowast120591119900 120595

1015840=

119908lowast120595

V21

(119906ℓ1015840

1 119906ℓ1015840

2) =

119908lowast

V1

(119906ℓ

1 119906ℓ

2) 119901

ℓ1015840

=

119901ℓ

120582ℓ 119905

ℓ1015840

22=

119905ℓ

22

120582ℓ

(12)

where 119908lowast = 1198621198901198621111989611 V21= 11986211120588 and 119908lowast is the character-

istic frequency of the medium

4 Solution of the Problem

We assume the solutions of the form

(1199061 1199062 119879 120595) = (1119882 119877 119878) 119880 exp 119894120585 (119909

1+ 1198981199092minus 119888119905) (13)

where 119888 = 120596120585 is the nondimensional phase velocity 120596 isthe frequency and 120585 is the wave number 119898 is still unknownparameter 1119882 119877 119878 are respectively the amplitude ratios of1199061 1199062 119879 and 119906ℓ

2with respect to 119906

1 Substituting the values of

1199061 1199062and 119879 from (13) in (8) (9) and (10) we obtain

[

119898

11986211

(11986212+ 119862119900minus

119875

2

)]119882 + (

minus1

119894120585

)119877

+ [1 minus 1198882+

1

11986211

(119862119900+

119875

2

)1198982] = 0

[

11986222

11986211

1198982+

1

11986211

(119862119900+

119875

2

) minus 1198882]119882

+ (

minus119898120573

119894120585

)119877 + [

1

11986211

(11986212+ 119862119900minus

119875

2

)119898] = 0

[1205761120573 (minus119898119888 + 119898119888

2120591119900119894120585)]119882 + [1 + 119896119898

2+ (

119888

119894120585

minus 1205911199001198882)]119877

+ 1205761[119888 minus 120591

1199001198882119894120585] = 0

(14)

where

1205761=

1198791199001205732

11

120588119908lowast11989611

(15)

The system of (14) has a nontrivial solution if the deter-minant of the coefficients [1119882 119877]119879 vanishes which yields tothe following polynomial characteristic equation

1198986+ 119889211198984+ 119889311198982+ 11988941= 0 (16)

where

1198891= [119896 (119871

21198921)]

1198892= [minus 119892

11198651198712+ 119896 (119871

11198921+ 11987121198922minus 1198602

1)

+ 11987121198921+ (120573)

2

12057611198651198712]

1198893= [minus 119865119892

1(1198711+ 1205761) minus 119865119892

21198712

+ 1205731198651198601(1 minus 120576

1) + 119896 (119871

11198922) + 11987111198921

+ 11987121198922+ (120573)

2

12057611198651198711+ 1198602

1(119865 minus 1)]

1198894= [minus119892

2119865 (1198711+ 1205761) + 11987111198922]

11988921=

1198892

1198891

11988931=

1198893

1198891

11988941=

1198894

1198891

119865 =

minus119888 + 1205911199001198882119894120585

119894120585

119860 = 1198981198601

1198601=

1

11986211

(11986212+ 119862119900minus

119875

2

)

119866 = 11989211198982+ 1198922 119892

1=

11986222

11986211

1198922=

1

11986211

(119862119900+

119875

2

) minus 1198882

119871 = 1198711+ 11987121198982 119871

1= 1 minus 119888

2

1198712=

1

11986211

(119862119900+

119875

2

)

(17)

The characteristic equation (16) is cubic in1198982 and hencepossesses three roots (1198982

119901 119901 = 1 2 3) Therefore there exist

three types of quasi-waves in transversely isotropic elastichalf-space namely quasi-longitudinal waves (QP) quasi-transverse waves (QSV) and quasi-thermal waves (QT)

The formal expression for displacements and volumefraction field satisfying the radiation condition that Re(119898

119901)ge

0 can be written as

(1199061 1199062 119879) =

3

sum

119901=1

119866119901(1 1198991119901 1198992119901) exp 119894120585 (119909

1+ 1198941198981199011199092minus 119888119905)

(18)

where

1198991119901=

Δ1

Δ

1198992119901=

Δ2

Δ

Δ1= (

minus1205731198981205761119888 (1 minus 120591

119900119888119894120585)

119894120585

minus 119860119884)

Δ2= (119866120576

1119888 (1 minus 120591

119900119888119894120585) minus 119860119885)

Journal of Thermodynamics 5

Δ = (119866119884 +

120573119898119885

119894120585

)

119884 = 1 + 1198961198982+ (

119888

119894120585

minus 1205911199001198882)

119885 = 1205761120573119898119888 (minus1 + 119888120591

119900119894120585)

(19)

and 1198661 1198662 1198663are arbitrary constants

Substituting the values of 120595 from (13) in (5) and aftersimplification we obtain

120595 = [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)] exp (119894120585 (119909

1minus 119888119905))

(20)

where

1198984= plusmn((

V1119888

120572ℓ)

2

minus 1)

12

(21)

and 1198664 1198665are arbitrary constants

The normal displacement and normal stress for medium1198722 with the help of (5) (6) (12) and (20) are given by

119906ℓ

2= 1198984120585 [1198664cos (120585119898

41199092) minus 1198665sin (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

119905ℓ

22= minus 120585

2(1 + 119898

2

4) [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

(22)

Also from (3) we have

11990522= 1198621211990611+ 1198622211990622minus 12057322119879

11990521= 119862119900(11990612+ 11990621)

(23)

5 Boundary Conditions

The appropriate boundary conditions are as follows

(i) vanishing of normal stress component at the freesurface of the liquid layer 119909

2= minus119867 that is

(119905ℓ

22)1198722

= minus119901ℓ= 0 (24)

(ii) continuity of normal stress components at the inter-face 119909

2= 0 that is

(11990522)1198721

= minus119901ℓ (25)

(iii) normal displacement components at the interface1199092= 0 that is

(119906ℓ

2)1198722

= (2)1198721

(26)

(iv) vanishing shear stress for medium1198721at the interface

1199092= 0 that is

(11990521)1198721

= 0 (27)

(v) thermal boundary conditions

120597119879

1205971199092

+ ℎ119879 = 0 (28)

where ℎ 997888rarr 0 corresponds Insulated boundarycondition ℎ 997888rarr infin corresponds Isolated boundarycondition

6 Derivations of the Secular Equations

Substituting the value of 1199062 119879 119906ℓ

2 119905ℓ

22from (18) and (22) in

(24)ndash(28) andwith the aid of (7) and (12) after simplificationwe obtain

1198882= (

120572ℓ

V1

)

2

[

[

(tanminus1 (11987911))

2

+ 12058521198672

12058521198672

]

]

(29)

where

11987911=

119862111198984(119864111198631minus 119864121198632+ 119864131198633)

119894119888120588ℓV21(119899111198631minus 119899121198632+ 119899131198633)

1198641119901= 119894120585 (

11986212

11986211

) + (

11986222

11986211

)1198991119901(1198942120585119898119901) minus (

12057322

12057311

)1198992119901

1198642119901= 120585 (119899

1119901119894 minus 119898119901) 119864

3119901= 1198992119901(ℎ minus 120585119898

119901)

(119901 = 1 2 3)

1198631= 1198642211986433minus 1198643211986423 119863

2= 1198642111986433minus 1198643111986423

1198633= 1198642111986432minus 1198643111986422

(30)

Equation (29) is surface waves secular equation of fiber-reinforced transversely isotropic thermoelastic half-spaceunder a uniform homogeneous inviscid liquid layer Thesecular equation has complete information about the phasevelocity wave number and attenuation coefficient of surfacewaves in such medium

61 Particular Cases

(i) If we take ℎ rarr 0 in (29) we obtain the frequencyequation for the thermally insulated boundary

(ii) If we take ℎ rarr infin in (29) we obtain the frequencyequation for the thermally isolated boundary

(iii) Taking 119875 rarr 0 in secular equation (29) yield thefrequency equation for fiber-reinforced transverselyisotropic thermoelastic half space without initialstress under a uniform homogeneous inviscid liquidlayer

(iv) For 120583119879= 120583119871= 120583 119870

11= 11987022= 119870 120573

11= 12057322= 120573 =

(3120582 + 2120583)120572119905 (29) reduces to the frequency equation

in a homogeneous Isotropic half space under inviscidliquid

(v) 120591119900rarr 0 in (10) we obtained the frequency equation

in fiber-reinforced coupled thermoelastic half spaceunder inviscid liquid layer

6 Journal of Thermodynamics

7 Rayleigh Wave

(1) Taking119867 rarr 0 the frequency equation (29) reduces to

11986411(1198642211986433minus 1198642311986432) minus 11986412(1198642111986433minus 1198643111986423)

+ 11986413(1198642111986432minus 1198643111986422) = 0

(31)

Equation (31) is the frequency equation for Rayleighsurface wave propagation in a fiber-reinforced transverselyisotropic thermoelastic half-space with initial stress In gen-eral wave number and hence phase velocity of waves iscomplex quantity therefore the waves are attenuated in spaceIf we write

119888minus1= Vminus1 + 119894120596minus1119865 (32)

so that 120585 = 119870 + 119894119865 where119870 = 120596V V and 119865 are real Also theroots of characteristic equation (16) are in general complexand hence we assume that119898

119901= 119901119901+119894119902119901 so that the exponent

in the plane wave solutions (18) for the half-space becomes

119894119870 (1199091minus 119898119868

1199011199092minus V119905) minus 119870(

119865

119870

1199091+ 119898119877

1199011199092) (33)

where

119898119877

119901= 119901119901minus 119902119901

119865

119870

119898119868

119901= 119902119901+ 119901119901

119865

119870

(119901 = 1 2 3)

(34)

This shows that V is the propagation velocity and 119865 is theattenuation coefficient of wave

The equation (18) can be rewritten as

1199061=

3

sum

119901=1

119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

1199062=

3

sum

119901=1

1198991119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

119879 =

3

sum

119901=1

1198992119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

(35)

with 120582119901= 119870(119898

119877

119901+ 119894119898119868

119901) = 120582

119877

119901+ 119894120582119868

119901(say) (119901 = 1 2 3)

Moreover it is clear that

10038161003816100381610038161003816120582119877

119901

10038161003816100381610038161003816

2

minus

10038161003816100381610038161003816120582119868

119901

10038161003816100381610038161003816

2

= 1198702[(119898119877

119901)

2

minus (119898119868

119901)

2

] (36)

71 Normal Stress Normal stress at the interface 1199092= 0 is

given by

11990522= (119862

12119894120585

3

sum

119901=1

119866119901minus 11986222120585

3

sum

119901=1

1198981199011198661199011198991119901minus 12057322

3

sum

119901=1

1198661199011198992119901)

times exp (119894120585 (1199091minus V119905)) (119901 = 1 2 3)

(37)

72 Specific Loss The Specific loss is the ratio of energy (Δ119882)dissipated in taking a specimen through a stress cycle to theelastic energy (119882) stored in the specimen when the strainis maximum The Specific loss is the most direct method ofdefining internal friction for amaterial For a sinusoidal planewave of small amplitude Kolsky [32] shows that the specificloss Δ119882119882 equals 4120587 times the absolute value of imaginarypart of 120585 to the real part of 120585 that is

Δ119882

119882

= 4120587

10038161003816100381610038161003816100381610038161003816

Im (120585)

Re (120585)

10038161003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

V119866120596

1003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

119866

119865

1003816100381610038161003816100381610038161003816

(38)

8 Numerical Results and Discussion

For numerical computations we take the following valuesof the relevant parameters for generalized fiber-reinforcedtransversely isotropic thermoelastic solid

120582 = 566 times 1010Nm2 120588 = 2660Kgm3

120583119879= 246 times 10

10Nm2 120583119871= 566 times 10

10Nm2

120572 = minus128 times 1010Nm2 120573 = 22090 times 10

10Nm2

11987011= 00921 times 10

3 Jmminus1 degminus1 sminus1 119875 = 100

11987022= 00963 times 10

3 Jmminus1 degminus1 sminus1

1205721= 0017 times 10

4 degminus1

1205722= 0015 times 10

4 degminus1 119879119900= 293K

120591119900= 005 s ℎ = 1 119867 = 1

119862119890= 0787 times 10

3 JKgminus1degminus1 120596 = 2 sminus1(39)

The physical constants for water are given by Ewing et al[31]

120588ℓ= 10 times 10

3 Kgm3 120582ℓ= 214 times 10

3Nm2 (40)

For comparison with the generalized fiber-reinforcedisotropic thermoelastic solid following Ezzat [33] we take

Journal of Thermodynamics 7

0 1 2 3 4 5

Phas

e velo

city

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

25

20

15

10

05

00

times101

Figure 2 Variation of phase velocity wrt wave number (with andwithout initial stress)

the following values of relevant parameters for the case ofcopper material as

120582 = 776 times 1010Nm2

120583119879= 120583119871= 120583 = 386 times 10

10Nm2

11987011= 11987022= 119870 = 386NKminus1 sminus1

12057311= 12057322= 120573 = (3120582 + 2120583) 120572

119905

120572119905= 178 times 10

minus5 Kminus1

1205721= 1005 degminus1 120572

2= 1001 degminus1

119862119890= 3831m2 Kminus1 sminus1 119875 = 0

119867 = 1 120588 = 8954Kgm3

120591119900= 01 s ℎ = 1

(41)

Using the above values of parameters the dispersioncurves of nondimensional phase velocity attenuation coef-ficient and specific heat are shown graphically with respectto nondimensional wave number Here we consider abbre-viations FTTIWIS FTTIIS FTIWIS and FTIIS for fiber-reinforced transversely isotropic thermoelasticmaterial with-out initial stress fiber-reinforced transversely isotropic ther-moelastic material with initial stress fiber-reinforced iso-tropic thermoelastic material without initial stress and fiber-reinforced isotropic thermoelastic material with initial stressrespectively and119867 refers to the thickness of the layer

Figure 2 depicts the variations of phase velocity withrespect to the wave number The behavior of phase velocity

0 1 2 3 4 5Wave number

05

00

minus05

minus10

minus15

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times10minus1

Figure 3 Variation of attenuation coefficient wrt wave number(with and without initial stress)

for FTTIWIS FTTIIS FTIWIS and FTIIS is similar butthe corresponding values are different in magnitude Thevalues of phase velocity decrease for lower wave numberand become stable for higher wave number in all the casesFigure 3 shows the variation of attenuation coefficient withrespect to the wave number For FTTIWIS and FTTIIS thevalues of attenuation coefficient increase monotonically upto wave number (120585) = 2 and after that it becomes constant forwhole range of wave number On the other hand the values ofattenuation coefficients are consistent within the whole rangeof wave number for FTIWIS and FTIIS

Figure 4 indicates the variation of phase velocity withrespect to thewave number for different values of thickness oflayer Here we consider the particular case of fiber-reinforcedtransversely isotropic thermoelastic material and analyses ofthe behavior of curves with respect to initial stress It isobserved that for small value of wave number the graph ofphase velocity shows opposite behavior wrt thickness thatis on increasing the values of thickness of layer the values ofcorresponding phase velocity decreases and it became steadyfor large values of wave number

Figure 5 shows the variation of attenuation coefficientwith respect to the wave number for different value ofthickness of the layer Here also we considered the particularcase of fiber-reinforced transversely isotropic thermoelasticmaterial and analyses of the behavior of curveswith respect toinitial stress It is perceived that on decreasing the value of119867the curve shows consistency for small value of wave numberand on increasing the value of119867 the curve shows uniformityfor large value of 120585

Figure 6 depicts the variations of specific loss verseswave number For FTIWIS and FTIIS the curve shows sharpdecrease in the value of specific loss for the range 0 le 120585 le 1and become parallel for 120585 gt 1 It is observed that graphs

8 Journal of Thermodynamics

0 1 2 3 4 5Wave number

25

20

15

10

05

00

Phas

e velo

city

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

times102

Figure 4 Variation of phase velocity wrt wave number (with andwithout initial stress for different value of119867)

corresponding to FTTIWIS and FTTIIS follow the similarpattern for whole range of 120585 but the values are different inmagnitude

81 Rayleigh Wave Figure 7 shows the variations of phasevelocity of Rayleigh wave verses wave number for FTTIWISFTTIIS FTIWIS and FTIIS The value of phase velocitydecreases for small values of 120585 and become stable for highervalue of 120585 It is analyzed that in all the cases behavior of curvesare almost the same within the range 2 le 120585 le 5 Figure 8shows the variations of attenuation coefficient of Rayleighwave verses wave number for FTTIWIS FTTIIS FTIWISand FTIIS For FTIWIS and FTIIS the graph shows a suddenfall for 120585 = 05 and then increases monotonically up to 120585 = 2and becomes constant for the remaining range of 120585 A largedifference between the behavior of curves for the transverselyisotropic and isotropic case is also perceived

9 Conclusions

The propagation of surface waves in an initially stressedfiber-reinforced transversely isotropic thermoelastic mate-rial under an inviscid liquid layer has been studied Afterdeveloping the formal solutions the secular equation forsurface wave propagation is derived Some special casesof frequency equation are also discussed The attenuationcoefficients and specific loss (liquid layer and half-space)have also been studied and computed for FTTIWIS FTTIISFTIWIS and FTIISThe trend of phase velocity is the same inall these four cases for surface wave but varies for attenuation

0 1 2 3 4 5Wave number

00

05

times10minus1

minus15

minus05

minus10

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

Figure 5 Variation of attenuation coeff wrt wave number (withand without initial stress for different value of119867)

0 1 2 3 4 5

Spec

ific l

oss

40

35

30

25

20

15

10

05

00

minus05

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

times10minus4

Figure 6 Variation of specific loss wrt wave number (with andwithout initial stress)

coefficient From numerical analysis it is shown that thephase speeds and attenuation coefficients for Rayleigh waveare also affected significantly due to the presence of initialstress parameters

Journal of Thermodynamics 9

0 1 2 3 4 5

Phas

e velo

city

Wave number

12

10

80

times103

60

40

20

00

FTTIWISFTTIIS

FTIWISFTIIS

Figure 7 Variation of phase velocity wrt wave number forRayleigh wave (with and without initial stress)

0 1 2 3 4 5

00

minus20

minus40

minus60

minus80

minus10

Wave number

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times105

Figure 8 Variation of attenuation coeff wrt wave number forRayleigh wave (with and without initial stress)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A J M Spencer Deformation of Fibre-reinforced MaterialsUniversity of Oxford Clarendon Va USA 1941

[2] A J Belfield T G Rogers and A JM Spencer ldquoStress in elasticplates reinforced by fibres lying in concentric circlesrdquo Journal oftheMechanics and Physics of Solids vol 31 no 1 pp 25ndash54 1983

[3] A C Pipkin ldquoFinite deformations of ideal fiber-reinforcedcompositesrdquo in Composites Materials G P Sendeckyj Ed vol2 of Mechanics of Composite Materials pp 251ndash308 AcademicPress New York NY USA 1973

[4] P R Sengupta and S Nath ldquoSurface waves in fibre-reinforcedanisotropic elastic mediardquo Indian Academy of Sciences SadhanaProceedings vol 26 pp 363ndash370 2001

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

[6] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 198081

[7] A U Erdem ldquoHeat Conduction in fiber-reinforced rigid bod-iesrdquo 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK) 6ndash8Eylul Ankara Turkey 1995

[8] R Kumar and R R Gupta ldquoStudy of wave motion in ananisotropic fiber-reinforced thermoelastic solidrdquo Journal ofSolid Mechanics vol 2 no 1 pp 91ndash100 2010

[9] P Chadwick and L T C Seet ldquoWave propagation in atransversely isotropic heat-conducting elastic materialrdquo Math-ematika vol 17 pp 255ndash274 1970

[10] H Singh and J N Sharma ldquoGeneralized thermoelastic waves intransversely isotropicmediardquo Journal of the Acoustical Society ofAmerica vol 77 pp 1046ndash1053 1985

[11] B Singh ldquoWave propagation in an anisotropic generalizedthermoelastic solidrdquo Indian Journal of Pure and Applied Mathe-matics vol 34 no 10 pp 1479ndash1485 2003

[12] P Chadwick Progress in Solid Mechanics vol 1 North-HollandAmsterdam The Netherlands 1960 edited by R Hill I NSneddon

[13] P Chadwick and DWWindle ldquoPropagation of Rayleigh wavesalong isothermal and insulated boundariesrdquo Proceedings of theRoyal Society of America vol 280 pp 47ndash71 1964

[14] J N Sharma and H Singh ldquoThermoelastic surface waves ina transversely isotropic half space with thermal relaxationsrdquoIndian Journal of Pure and Applied Mathematics vol 16 no 10pp 1202ndash1219 1985

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

[16] J Wang and P Slattery ldquoThermoelasticity without energydissipation for initially stressed bodiesrdquo International Journal ofMathematics and Mathematical Sciences vol 31 no 6 pp 329ndash337 2002

[17] D Iesan ldquoA theory of prestressed thermoelastic Cosserat con-tinuardquo Journal of Applied Mathematics and Mechanics vol 88no 4 pp 306ndash319 2008

[18] K Ames and B Straughan ldquoContinuous dependence results forinitially prestressed thermoelastic bodiesrdquo International Journalof Engineering Science vol 30 no 1 pp 7ndash13 1992

[19] J Wang R S Dhaliwal and S R Majumdar ldquoSome theoremsin the generalized theory of thermoelasticity for prestressedbodiesrdquo Indian Journal of Pure and Applied Mathematics vol28 no 2 pp 267ndash276 1997

[20] M Marin and C Marinescu ldquoThermoelasticity of initiallystressed bodies asymptotic equipartition of energiesrdquo Interna-tional Journal of Engineering Science vol 36 no 1 pp 73ndash861998

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in Condensed Matter Physics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Thermodynamics 5

Δ = (119866119884 +

120573119898119885

119894120585

)

119884 = 1 + 1198961198982+ (

119888

119894120585

minus 1205911199001198882)

119885 = 1205761120573119898119888 (minus1 + 119888120591

119900119894120585)

(19)

and 1198661 1198662 1198663are arbitrary constants

Substituting the values of 120595 from (13) in (5) and aftersimplification we obtain

120595 = [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)] exp (119894120585 (119909

1minus 119888119905))

(20)

where

1198984= plusmn((

V1119888

120572ℓ)

2

minus 1)

12

(21)

and 1198664 1198665are arbitrary constants

The normal displacement and normal stress for medium1198722 with the help of (5) (6) (12) and (20) are given by

119906ℓ

2= 1198984120585 [1198664cos (120585119898

41199092) minus 1198665sin (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

119905ℓ

22= minus 120585

2(1 + 119898

2

4) [1198664sin (120585119898

41199092) + 1198665cos (120585119898

41199092)]

times exp (119894120585 (1199091minus 119888119905))

(22)

Also from (3) we have

11990522= 1198621211990611+ 1198622211990622minus 12057322119879

11990521= 119862119900(11990612+ 11990621)

(23)

5 Boundary Conditions

The appropriate boundary conditions are as follows

(i) vanishing of normal stress component at the freesurface of the liquid layer 119909

2= minus119867 that is

(119905ℓ

22)1198722

= minus119901ℓ= 0 (24)

(ii) continuity of normal stress components at the inter-face 119909

2= 0 that is

(11990522)1198721

= minus119901ℓ (25)

(iii) normal displacement components at the interface1199092= 0 that is

(119906ℓ

2)1198722

= (2)1198721

(26)

(iv) vanishing shear stress for medium1198721at the interface

1199092= 0 that is

(11990521)1198721

= 0 (27)

(v) thermal boundary conditions

120597119879

1205971199092

+ ℎ119879 = 0 (28)

where ℎ 997888rarr 0 corresponds Insulated boundarycondition ℎ 997888rarr infin corresponds Isolated boundarycondition

6 Derivations of the Secular Equations

Substituting the value of 1199062 119879 119906ℓ

2 119905ℓ

22from (18) and (22) in

(24)ndash(28) andwith the aid of (7) and (12) after simplificationwe obtain

1198882= (

120572ℓ

V1

)

2

[

[

(tanminus1 (11987911))

2

+ 12058521198672

12058521198672

]

]

(29)

where

11987911=

119862111198984(119864111198631minus 119864121198632+ 119864131198633)

119894119888120588ℓV21(119899111198631minus 119899121198632+ 119899131198633)

1198641119901= 119894120585 (

11986212

11986211

) + (

11986222

11986211

)1198991119901(1198942120585119898119901) minus (

12057322

12057311

)1198992119901

1198642119901= 120585 (119899

1119901119894 minus 119898119901) 119864

3119901= 1198992119901(ℎ minus 120585119898

119901)

(119901 = 1 2 3)

1198631= 1198642211986433minus 1198643211986423 119863

2= 1198642111986433minus 1198643111986423

1198633= 1198642111986432minus 1198643111986422

(30)

Equation (29) is surface waves secular equation of fiber-reinforced transversely isotropic thermoelastic half-spaceunder a uniform homogeneous inviscid liquid layer Thesecular equation has complete information about the phasevelocity wave number and attenuation coefficient of surfacewaves in such medium

61 Particular Cases

(i) If we take ℎ rarr 0 in (29) we obtain the frequencyequation for the thermally insulated boundary

(ii) If we take ℎ rarr infin in (29) we obtain the frequencyequation for the thermally isolated boundary

(iii) Taking 119875 rarr 0 in secular equation (29) yield thefrequency equation for fiber-reinforced transverselyisotropic thermoelastic half space without initialstress under a uniform homogeneous inviscid liquidlayer

(iv) For 120583119879= 120583119871= 120583 119870

11= 11987022= 119870 120573

11= 12057322= 120573 =

(3120582 + 2120583)120572119905 (29) reduces to the frequency equation

in a homogeneous Isotropic half space under inviscidliquid

(v) 120591119900rarr 0 in (10) we obtained the frequency equation

in fiber-reinforced coupled thermoelastic half spaceunder inviscid liquid layer

6 Journal of Thermodynamics

7 Rayleigh Wave

(1) Taking119867 rarr 0 the frequency equation (29) reduces to

11986411(1198642211986433minus 1198642311986432) minus 11986412(1198642111986433minus 1198643111986423)

+ 11986413(1198642111986432minus 1198643111986422) = 0

(31)

Equation (31) is the frequency equation for Rayleighsurface wave propagation in a fiber-reinforced transverselyisotropic thermoelastic half-space with initial stress In gen-eral wave number and hence phase velocity of waves iscomplex quantity therefore the waves are attenuated in spaceIf we write

119888minus1= Vminus1 + 119894120596minus1119865 (32)

so that 120585 = 119870 + 119894119865 where119870 = 120596V V and 119865 are real Also theroots of characteristic equation (16) are in general complexand hence we assume that119898

119901= 119901119901+119894119902119901 so that the exponent

in the plane wave solutions (18) for the half-space becomes

119894119870 (1199091minus 119898119868

1199011199092minus V119905) minus 119870(

119865

119870

1199091+ 119898119877

1199011199092) (33)

where

119898119877

119901= 119901119901minus 119902119901

119865

119870

119898119868

119901= 119902119901+ 119901119901

119865

119870

(119901 = 1 2 3)

(34)

This shows that V is the propagation velocity and 119865 is theattenuation coefficient of wave

The equation (18) can be rewritten as

1199061=

3

sum

119901=1

119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

1199062=

3

sum

119901=1

1198991119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

119879 =

3

sum

119901=1

1198992119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

(35)

with 120582119901= 119870(119898

119877

119901+ 119894119898119868

119901) = 120582

119877

119901+ 119894120582119868

119901(say) (119901 = 1 2 3)

Moreover it is clear that

10038161003816100381610038161003816120582119877

119901

10038161003816100381610038161003816

2

minus

10038161003816100381610038161003816120582119868

119901

10038161003816100381610038161003816

2

= 1198702[(119898119877

119901)

2

minus (119898119868

119901)

2

] (36)

71 Normal Stress Normal stress at the interface 1199092= 0 is

given by

11990522= (119862

12119894120585

3

sum

119901=1

119866119901minus 11986222120585

3

sum

119901=1

1198981199011198661199011198991119901minus 12057322

3

sum

119901=1

1198661199011198992119901)

times exp (119894120585 (1199091minus V119905)) (119901 = 1 2 3)

(37)

72 Specific Loss The Specific loss is the ratio of energy (Δ119882)dissipated in taking a specimen through a stress cycle to theelastic energy (119882) stored in the specimen when the strainis maximum The Specific loss is the most direct method ofdefining internal friction for amaterial For a sinusoidal planewave of small amplitude Kolsky [32] shows that the specificloss Δ119882119882 equals 4120587 times the absolute value of imaginarypart of 120585 to the real part of 120585 that is

Δ119882

119882

= 4120587

10038161003816100381610038161003816100381610038161003816

Im (120585)

Re (120585)

10038161003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

V119866120596

1003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

119866

119865

1003816100381610038161003816100381610038161003816

(38)

8 Numerical Results and Discussion

For numerical computations we take the following valuesof the relevant parameters for generalized fiber-reinforcedtransversely isotropic thermoelastic solid

120582 = 566 times 1010Nm2 120588 = 2660Kgm3

120583119879= 246 times 10

10Nm2 120583119871= 566 times 10

10Nm2

120572 = minus128 times 1010Nm2 120573 = 22090 times 10

10Nm2

11987011= 00921 times 10

3 Jmminus1 degminus1 sminus1 119875 = 100

11987022= 00963 times 10

3 Jmminus1 degminus1 sminus1

1205721= 0017 times 10

4 degminus1

1205722= 0015 times 10

4 degminus1 119879119900= 293K

120591119900= 005 s ℎ = 1 119867 = 1

119862119890= 0787 times 10

3 JKgminus1degminus1 120596 = 2 sminus1(39)

The physical constants for water are given by Ewing et al[31]

120588ℓ= 10 times 10

3 Kgm3 120582ℓ= 214 times 10

3Nm2 (40)

For comparison with the generalized fiber-reinforcedisotropic thermoelastic solid following Ezzat [33] we take

Journal of Thermodynamics 7

0 1 2 3 4 5

Phas

e velo

city

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

25

20

15

10

05

00

times101

Figure 2 Variation of phase velocity wrt wave number (with andwithout initial stress)

the following values of relevant parameters for the case ofcopper material as

120582 = 776 times 1010Nm2

120583119879= 120583119871= 120583 = 386 times 10

10Nm2

11987011= 11987022= 119870 = 386NKminus1 sminus1

12057311= 12057322= 120573 = (3120582 + 2120583) 120572

119905

120572119905= 178 times 10

minus5 Kminus1

1205721= 1005 degminus1 120572

2= 1001 degminus1

119862119890= 3831m2 Kminus1 sminus1 119875 = 0

119867 = 1 120588 = 8954Kgm3

120591119900= 01 s ℎ = 1

(41)

Using the above values of parameters the dispersioncurves of nondimensional phase velocity attenuation coef-ficient and specific heat are shown graphically with respectto nondimensional wave number Here we consider abbre-viations FTTIWIS FTTIIS FTIWIS and FTIIS for fiber-reinforced transversely isotropic thermoelasticmaterial with-out initial stress fiber-reinforced transversely isotropic ther-moelastic material with initial stress fiber-reinforced iso-tropic thermoelastic material without initial stress and fiber-reinforced isotropic thermoelastic material with initial stressrespectively and119867 refers to the thickness of the layer

Figure 2 depicts the variations of phase velocity withrespect to the wave number The behavior of phase velocity

0 1 2 3 4 5Wave number

05

00

minus05

minus10

minus15

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times10minus1

Figure 3 Variation of attenuation coefficient wrt wave number(with and without initial stress)

for FTTIWIS FTTIIS FTIWIS and FTIIS is similar butthe corresponding values are different in magnitude Thevalues of phase velocity decrease for lower wave numberand become stable for higher wave number in all the casesFigure 3 shows the variation of attenuation coefficient withrespect to the wave number For FTTIWIS and FTTIIS thevalues of attenuation coefficient increase monotonically upto wave number (120585) = 2 and after that it becomes constant forwhole range of wave number On the other hand the values ofattenuation coefficients are consistent within the whole rangeof wave number for FTIWIS and FTIIS

Figure 4 indicates the variation of phase velocity withrespect to thewave number for different values of thickness oflayer Here we consider the particular case of fiber-reinforcedtransversely isotropic thermoelastic material and analyses ofthe behavior of curves with respect to initial stress It isobserved that for small value of wave number the graph ofphase velocity shows opposite behavior wrt thickness thatis on increasing the values of thickness of layer the values ofcorresponding phase velocity decreases and it became steadyfor large values of wave number

Figure 5 shows the variation of attenuation coefficientwith respect to the wave number for different value ofthickness of the layer Here also we considered the particularcase of fiber-reinforced transversely isotropic thermoelasticmaterial and analyses of the behavior of curveswith respect toinitial stress It is perceived that on decreasing the value of119867the curve shows consistency for small value of wave numberand on increasing the value of119867 the curve shows uniformityfor large value of 120585

Figure 6 depicts the variations of specific loss verseswave number For FTIWIS and FTIIS the curve shows sharpdecrease in the value of specific loss for the range 0 le 120585 le 1and become parallel for 120585 gt 1 It is observed that graphs

8 Journal of Thermodynamics

0 1 2 3 4 5Wave number

25

20

15

10

05

00

Phas

e velo

city

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

times102

Figure 4 Variation of phase velocity wrt wave number (with andwithout initial stress for different value of119867)

corresponding to FTTIWIS and FTTIIS follow the similarpattern for whole range of 120585 but the values are different inmagnitude

81 Rayleigh Wave Figure 7 shows the variations of phasevelocity of Rayleigh wave verses wave number for FTTIWISFTTIIS FTIWIS and FTIIS The value of phase velocitydecreases for small values of 120585 and become stable for highervalue of 120585 It is analyzed that in all the cases behavior of curvesare almost the same within the range 2 le 120585 le 5 Figure 8shows the variations of attenuation coefficient of Rayleighwave verses wave number for FTTIWIS FTTIIS FTIWISand FTIIS For FTIWIS and FTIIS the graph shows a suddenfall for 120585 = 05 and then increases monotonically up to 120585 = 2and becomes constant for the remaining range of 120585 A largedifference between the behavior of curves for the transverselyisotropic and isotropic case is also perceived

9 Conclusions

The propagation of surface waves in an initially stressedfiber-reinforced transversely isotropic thermoelastic mate-rial under an inviscid liquid layer has been studied Afterdeveloping the formal solutions the secular equation forsurface wave propagation is derived Some special casesof frequency equation are also discussed The attenuationcoefficients and specific loss (liquid layer and half-space)have also been studied and computed for FTTIWIS FTTIISFTIWIS and FTIISThe trend of phase velocity is the same inall these four cases for surface wave but varies for attenuation

0 1 2 3 4 5Wave number

00

05

times10minus1

minus15

minus05

minus10

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

Figure 5 Variation of attenuation coeff wrt wave number (withand without initial stress for different value of119867)

0 1 2 3 4 5

Spec

ific l

oss

40

35

30

25

20

15

10

05

00

minus05

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

times10minus4

Figure 6 Variation of specific loss wrt wave number (with andwithout initial stress)

coefficient From numerical analysis it is shown that thephase speeds and attenuation coefficients for Rayleigh waveare also affected significantly due to the presence of initialstress parameters

Journal of Thermodynamics 9

0 1 2 3 4 5

Phas

e velo

city

Wave number

12

10

80

times103

60

40

20

00

FTTIWISFTTIIS

FTIWISFTIIS

Figure 7 Variation of phase velocity wrt wave number forRayleigh wave (with and without initial stress)

0 1 2 3 4 5

00

minus20

minus40

minus60

minus80

minus10

Wave number

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times105

Figure 8 Variation of attenuation coeff wrt wave number forRayleigh wave (with and without initial stress)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A J M Spencer Deformation of Fibre-reinforced MaterialsUniversity of Oxford Clarendon Va USA 1941

[2] A J Belfield T G Rogers and A JM Spencer ldquoStress in elasticplates reinforced by fibres lying in concentric circlesrdquo Journal oftheMechanics and Physics of Solids vol 31 no 1 pp 25ndash54 1983

[3] A C Pipkin ldquoFinite deformations of ideal fiber-reinforcedcompositesrdquo in Composites Materials G P Sendeckyj Ed vol2 of Mechanics of Composite Materials pp 251ndash308 AcademicPress New York NY USA 1973

[4] P R Sengupta and S Nath ldquoSurface waves in fibre-reinforcedanisotropic elastic mediardquo Indian Academy of Sciences SadhanaProceedings vol 26 pp 363ndash370 2001

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

[6] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 198081

[7] A U Erdem ldquoHeat Conduction in fiber-reinforced rigid bod-iesrdquo 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK) 6ndash8Eylul Ankara Turkey 1995

[8] R Kumar and R R Gupta ldquoStudy of wave motion in ananisotropic fiber-reinforced thermoelastic solidrdquo Journal ofSolid Mechanics vol 2 no 1 pp 91ndash100 2010

[9] P Chadwick and L T C Seet ldquoWave propagation in atransversely isotropic heat-conducting elastic materialrdquo Math-ematika vol 17 pp 255ndash274 1970

[10] H Singh and J N Sharma ldquoGeneralized thermoelastic waves intransversely isotropicmediardquo Journal of the Acoustical Society ofAmerica vol 77 pp 1046ndash1053 1985

[11] B Singh ldquoWave propagation in an anisotropic generalizedthermoelastic solidrdquo Indian Journal of Pure and Applied Mathe-matics vol 34 no 10 pp 1479ndash1485 2003

[12] P Chadwick Progress in Solid Mechanics vol 1 North-HollandAmsterdam The Netherlands 1960 edited by R Hill I NSneddon

[13] P Chadwick and DWWindle ldquoPropagation of Rayleigh wavesalong isothermal and insulated boundariesrdquo Proceedings of theRoyal Society of America vol 280 pp 47ndash71 1964

[14] J N Sharma and H Singh ldquoThermoelastic surface waves ina transversely isotropic half space with thermal relaxationsrdquoIndian Journal of Pure and Applied Mathematics vol 16 no 10pp 1202ndash1219 1985

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

[16] J Wang and P Slattery ldquoThermoelasticity without energydissipation for initially stressed bodiesrdquo International Journal ofMathematics and Mathematical Sciences vol 31 no 6 pp 329ndash337 2002

[17] D Iesan ldquoA theory of prestressed thermoelastic Cosserat con-tinuardquo Journal of Applied Mathematics and Mechanics vol 88no 4 pp 306ndash319 2008

[18] K Ames and B Straughan ldquoContinuous dependence results forinitially prestressed thermoelastic bodiesrdquo International Journalof Engineering Science vol 30 no 1 pp 7ndash13 1992

[19] J Wang R S Dhaliwal and S R Majumdar ldquoSome theoremsin the generalized theory of thermoelasticity for prestressedbodiesrdquo Indian Journal of Pure and Applied Mathematics vol28 no 2 pp 267ndash276 1997

[20] M Marin and C Marinescu ldquoThermoelasticity of initiallystressed bodies asymptotic equipartition of energiesrdquo Interna-tional Journal of Engineering Science vol 36 no 1 pp 73ndash861998

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

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Statistical MechanicsInternational Journal of

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AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Journal of Thermodynamics

7 Rayleigh Wave

(1) Taking119867 rarr 0 the frequency equation (29) reduces to

11986411(1198642211986433minus 1198642311986432) minus 11986412(1198642111986433minus 1198643111986423)

+ 11986413(1198642111986432minus 1198643111986422) = 0

(31)

Equation (31) is the frequency equation for Rayleighsurface wave propagation in a fiber-reinforced transverselyisotropic thermoelastic half-space with initial stress In gen-eral wave number and hence phase velocity of waves iscomplex quantity therefore the waves are attenuated in spaceIf we write

119888minus1= Vminus1 + 119894120596minus1119865 (32)

so that 120585 = 119870 + 119894119865 where119870 = 120596V V and 119865 are real Also theroots of characteristic equation (16) are in general complexand hence we assume that119898

119901= 119901119901+119894119902119901 so that the exponent

in the plane wave solutions (18) for the half-space becomes

119894119870 (1199091minus 119898119868

1199011199092minus V119905) minus 119870(

119865

119870

1199091+ 119898119877

1199011199092) (33)

where

119898119877

119901= 119901119901minus 119902119901

119865

119870

119898119868

119901= 119902119901+ 119901119901

119865

119870

(119901 = 1 2 3)

(34)

This shows that V is the propagation velocity and 119865 is theattenuation coefficient of wave

The equation (18) can be rewritten as

1199061=

3

sum

119901=1

119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

1199062=

3

sum

119901=1

1198991119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

119879 =

3

sum

119901=1

1198992119901119866119901exp (minus119865119909

1minus 120582119877

1199011199092)

times exp [119894 (119870 (1199091minus V119905) minus 120582119868

1199011199092)]

(35)

with 120582119901= 119870(119898

119877

119901+ 119894119898119868

119901) = 120582

119877

119901+ 119894120582119868

119901(say) (119901 = 1 2 3)

Moreover it is clear that

10038161003816100381610038161003816120582119877

119901

10038161003816100381610038161003816

2

minus

10038161003816100381610038161003816120582119868

119901

10038161003816100381610038161003816

2

= 1198702[(119898119877

119901)

2

minus (119898119868

119901)

2

] (36)

71 Normal Stress Normal stress at the interface 1199092= 0 is

given by

11990522= (119862

12119894120585

3

sum

119901=1

119866119901minus 11986222120585

3

sum

119901=1

1198981199011198661199011198991119901minus 12057322

3

sum

119901=1

1198661199011198992119901)

times exp (119894120585 (1199091minus V119905)) (119901 = 1 2 3)

(37)

72 Specific Loss The Specific loss is the ratio of energy (Δ119882)dissipated in taking a specimen through a stress cycle to theelastic energy (119882) stored in the specimen when the strainis maximum The Specific loss is the most direct method ofdefining internal friction for amaterial For a sinusoidal planewave of small amplitude Kolsky [32] shows that the specificloss Δ119882119882 equals 4120587 times the absolute value of imaginarypart of 120585 to the real part of 120585 that is

Δ119882

119882

= 4120587

10038161003816100381610038161003816100381610038161003816

Im (120585)

Re (120585)

10038161003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

V119866120596

1003816100381610038161003816100381610038161003816

= 4120587

1003816100381610038161003816100381610038161003816

119866

119865

1003816100381610038161003816100381610038161003816

(38)

8 Numerical Results and Discussion

For numerical computations we take the following valuesof the relevant parameters for generalized fiber-reinforcedtransversely isotropic thermoelastic solid

120582 = 566 times 1010Nm2 120588 = 2660Kgm3

120583119879= 246 times 10

10Nm2 120583119871= 566 times 10

10Nm2

120572 = minus128 times 1010Nm2 120573 = 22090 times 10

10Nm2

11987011= 00921 times 10

3 Jmminus1 degminus1 sminus1 119875 = 100

11987022= 00963 times 10

3 Jmminus1 degminus1 sminus1

1205721= 0017 times 10

4 degminus1

1205722= 0015 times 10

4 degminus1 119879119900= 293K

120591119900= 005 s ℎ = 1 119867 = 1

119862119890= 0787 times 10

3 JKgminus1degminus1 120596 = 2 sminus1(39)

The physical constants for water are given by Ewing et al[31]

120588ℓ= 10 times 10

3 Kgm3 120582ℓ= 214 times 10

3Nm2 (40)

For comparison with the generalized fiber-reinforcedisotropic thermoelastic solid following Ezzat [33] we take

Journal of Thermodynamics 7

0 1 2 3 4 5

Phas

e velo

city

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

25

20

15

10

05

00

times101

Figure 2 Variation of phase velocity wrt wave number (with andwithout initial stress)

the following values of relevant parameters for the case ofcopper material as

120582 = 776 times 1010Nm2

120583119879= 120583119871= 120583 = 386 times 10

10Nm2

11987011= 11987022= 119870 = 386NKminus1 sminus1

12057311= 12057322= 120573 = (3120582 + 2120583) 120572

119905

120572119905= 178 times 10

minus5 Kminus1

1205721= 1005 degminus1 120572

2= 1001 degminus1

119862119890= 3831m2 Kminus1 sminus1 119875 = 0

119867 = 1 120588 = 8954Kgm3

120591119900= 01 s ℎ = 1

(41)

Using the above values of parameters the dispersioncurves of nondimensional phase velocity attenuation coef-ficient and specific heat are shown graphically with respectto nondimensional wave number Here we consider abbre-viations FTTIWIS FTTIIS FTIWIS and FTIIS for fiber-reinforced transversely isotropic thermoelasticmaterial with-out initial stress fiber-reinforced transversely isotropic ther-moelastic material with initial stress fiber-reinforced iso-tropic thermoelastic material without initial stress and fiber-reinforced isotropic thermoelastic material with initial stressrespectively and119867 refers to the thickness of the layer

Figure 2 depicts the variations of phase velocity withrespect to the wave number The behavior of phase velocity

0 1 2 3 4 5Wave number

05

00

minus05

minus10

minus15

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times10minus1

Figure 3 Variation of attenuation coefficient wrt wave number(with and without initial stress)

for FTTIWIS FTTIIS FTIWIS and FTIIS is similar butthe corresponding values are different in magnitude Thevalues of phase velocity decrease for lower wave numberand become stable for higher wave number in all the casesFigure 3 shows the variation of attenuation coefficient withrespect to the wave number For FTTIWIS and FTTIIS thevalues of attenuation coefficient increase monotonically upto wave number (120585) = 2 and after that it becomes constant forwhole range of wave number On the other hand the values ofattenuation coefficients are consistent within the whole rangeof wave number for FTIWIS and FTIIS

Figure 4 indicates the variation of phase velocity withrespect to thewave number for different values of thickness oflayer Here we consider the particular case of fiber-reinforcedtransversely isotropic thermoelastic material and analyses ofthe behavior of curves with respect to initial stress It isobserved that for small value of wave number the graph ofphase velocity shows opposite behavior wrt thickness thatis on increasing the values of thickness of layer the values ofcorresponding phase velocity decreases and it became steadyfor large values of wave number

Figure 5 shows the variation of attenuation coefficientwith respect to the wave number for different value ofthickness of the layer Here also we considered the particularcase of fiber-reinforced transversely isotropic thermoelasticmaterial and analyses of the behavior of curveswith respect toinitial stress It is perceived that on decreasing the value of119867the curve shows consistency for small value of wave numberand on increasing the value of119867 the curve shows uniformityfor large value of 120585

Figure 6 depicts the variations of specific loss verseswave number For FTIWIS and FTIIS the curve shows sharpdecrease in the value of specific loss for the range 0 le 120585 le 1and become parallel for 120585 gt 1 It is observed that graphs

8 Journal of Thermodynamics

0 1 2 3 4 5Wave number

25

20

15

10

05

00

Phas

e velo

city

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

times102

Figure 4 Variation of phase velocity wrt wave number (with andwithout initial stress for different value of119867)

corresponding to FTTIWIS and FTTIIS follow the similarpattern for whole range of 120585 but the values are different inmagnitude

81 Rayleigh Wave Figure 7 shows the variations of phasevelocity of Rayleigh wave verses wave number for FTTIWISFTTIIS FTIWIS and FTIIS The value of phase velocitydecreases for small values of 120585 and become stable for highervalue of 120585 It is analyzed that in all the cases behavior of curvesare almost the same within the range 2 le 120585 le 5 Figure 8shows the variations of attenuation coefficient of Rayleighwave verses wave number for FTTIWIS FTTIIS FTIWISand FTIIS For FTIWIS and FTIIS the graph shows a suddenfall for 120585 = 05 and then increases monotonically up to 120585 = 2and becomes constant for the remaining range of 120585 A largedifference between the behavior of curves for the transverselyisotropic and isotropic case is also perceived

9 Conclusions

The propagation of surface waves in an initially stressedfiber-reinforced transversely isotropic thermoelastic mate-rial under an inviscid liquid layer has been studied Afterdeveloping the formal solutions the secular equation forsurface wave propagation is derived Some special casesof frequency equation are also discussed The attenuationcoefficients and specific loss (liquid layer and half-space)have also been studied and computed for FTTIWIS FTTIISFTIWIS and FTIISThe trend of phase velocity is the same inall these four cases for surface wave but varies for attenuation

0 1 2 3 4 5Wave number

00

05

times10minus1

minus15

minus05

minus10

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

Figure 5 Variation of attenuation coeff wrt wave number (withand without initial stress for different value of119867)

0 1 2 3 4 5

Spec

ific l

oss

40

35

30

25

20

15

10

05

00

minus05

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

times10minus4

Figure 6 Variation of specific loss wrt wave number (with andwithout initial stress)

coefficient From numerical analysis it is shown that thephase speeds and attenuation coefficients for Rayleigh waveare also affected significantly due to the presence of initialstress parameters

Journal of Thermodynamics 9

0 1 2 3 4 5

Phas

e velo

city

Wave number

12

10

80

times103

60

40

20

00

FTTIWISFTTIIS

FTIWISFTIIS

Figure 7 Variation of phase velocity wrt wave number forRayleigh wave (with and without initial stress)

0 1 2 3 4 5

00

minus20

minus40

minus60

minus80

minus10

Wave number

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times105

Figure 8 Variation of attenuation coeff wrt wave number forRayleigh wave (with and without initial stress)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A J M Spencer Deformation of Fibre-reinforced MaterialsUniversity of Oxford Clarendon Va USA 1941

[2] A J Belfield T G Rogers and A JM Spencer ldquoStress in elasticplates reinforced by fibres lying in concentric circlesrdquo Journal oftheMechanics and Physics of Solids vol 31 no 1 pp 25ndash54 1983

[3] A C Pipkin ldquoFinite deformations of ideal fiber-reinforcedcompositesrdquo in Composites Materials G P Sendeckyj Ed vol2 of Mechanics of Composite Materials pp 251ndash308 AcademicPress New York NY USA 1973

[4] P R Sengupta and S Nath ldquoSurface waves in fibre-reinforcedanisotropic elastic mediardquo Indian Academy of Sciences SadhanaProceedings vol 26 pp 363ndash370 2001

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

[6] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 198081

[7] A U Erdem ldquoHeat Conduction in fiber-reinforced rigid bod-iesrdquo 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK) 6ndash8Eylul Ankara Turkey 1995

[8] R Kumar and R R Gupta ldquoStudy of wave motion in ananisotropic fiber-reinforced thermoelastic solidrdquo Journal ofSolid Mechanics vol 2 no 1 pp 91ndash100 2010

[9] P Chadwick and L T C Seet ldquoWave propagation in atransversely isotropic heat-conducting elastic materialrdquo Math-ematika vol 17 pp 255ndash274 1970

[10] H Singh and J N Sharma ldquoGeneralized thermoelastic waves intransversely isotropicmediardquo Journal of the Acoustical Society ofAmerica vol 77 pp 1046ndash1053 1985

[11] B Singh ldquoWave propagation in an anisotropic generalizedthermoelastic solidrdquo Indian Journal of Pure and Applied Mathe-matics vol 34 no 10 pp 1479ndash1485 2003

[12] P Chadwick Progress in Solid Mechanics vol 1 North-HollandAmsterdam The Netherlands 1960 edited by R Hill I NSneddon

[13] P Chadwick and DWWindle ldquoPropagation of Rayleigh wavesalong isothermal and insulated boundariesrdquo Proceedings of theRoyal Society of America vol 280 pp 47ndash71 1964

[14] J N Sharma and H Singh ldquoThermoelastic surface waves ina transversely isotropic half space with thermal relaxationsrdquoIndian Journal of Pure and Applied Mathematics vol 16 no 10pp 1202ndash1219 1985

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

[16] J Wang and P Slattery ldquoThermoelasticity without energydissipation for initially stressed bodiesrdquo International Journal ofMathematics and Mathematical Sciences vol 31 no 6 pp 329ndash337 2002

[17] D Iesan ldquoA theory of prestressed thermoelastic Cosserat con-tinuardquo Journal of Applied Mathematics and Mechanics vol 88no 4 pp 306ndash319 2008

[18] K Ames and B Straughan ldquoContinuous dependence results forinitially prestressed thermoelastic bodiesrdquo International Journalof Engineering Science vol 30 no 1 pp 7ndash13 1992

[19] J Wang R S Dhaliwal and S R Majumdar ldquoSome theoremsin the generalized theory of thermoelasticity for prestressedbodiesrdquo Indian Journal of Pure and Applied Mathematics vol28 no 2 pp 267ndash276 1997

[20] M Marin and C Marinescu ldquoThermoelasticity of initiallystressed bodies asymptotic equipartition of energiesrdquo Interna-tional Journal of Engineering Science vol 36 no 1 pp 73ndash861998

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Thermodynamics 7

0 1 2 3 4 5

Phas

e velo

city

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

25

20

15

10

05

00

times101

Figure 2 Variation of phase velocity wrt wave number (with andwithout initial stress)

the following values of relevant parameters for the case ofcopper material as

120582 = 776 times 1010Nm2

120583119879= 120583119871= 120583 = 386 times 10

10Nm2

11987011= 11987022= 119870 = 386NKminus1 sminus1

12057311= 12057322= 120573 = (3120582 + 2120583) 120572

119905

120572119905= 178 times 10

minus5 Kminus1

1205721= 1005 degminus1 120572

2= 1001 degminus1

119862119890= 3831m2 Kminus1 sminus1 119875 = 0

119867 = 1 120588 = 8954Kgm3

120591119900= 01 s ℎ = 1

(41)

Using the above values of parameters the dispersioncurves of nondimensional phase velocity attenuation coef-ficient and specific heat are shown graphically with respectto nondimensional wave number Here we consider abbre-viations FTTIWIS FTTIIS FTIWIS and FTIIS for fiber-reinforced transversely isotropic thermoelasticmaterial with-out initial stress fiber-reinforced transversely isotropic ther-moelastic material with initial stress fiber-reinforced iso-tropic thermoelastic material without initial stress and fiber-reinforced isotropic thermoelastic material with initial stressrespectively and119867 refers to the thickness of the layer

Figure 2 depicts the variations of phase velocity withrespect to the wave number The behavior of phase velocity

0 1 2 3 4 5Wave number

05

00

minus05

minus10

minus15

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times10minus1

Figure 3 Variation of attenuation coefficient wrt wave number(with and without initial stress)

for FTTIWIS FTTIIS FTIWIS and FTIIS is similar butthe corresponding values are different in magnitude Thevalues of phase velocity decrease for lower wave numberand become stable for higher wave number in all the casesFigure 3 shows the variation of attenuation coefficient withrespect to the wave number For FTTIWIS and FTTIIS thevalues of attenuation coefficient increase monotonically upto wave number (120585) = 2 and after that it becomes constant forwhole range of wave number On the other hand the values ofattenuation coefficients are consistent within the whole rangeof wave number for FTIWIS and FTIIS

Figure 4 indicates the variation of phase velocity withrespect to thewave number for different values of thickness oflayer Here we consider the particular case of fiber-reinforcedtransversely isotropic thermoelastic material and analyses ofthe behavior of curves with respect to initial stress It isobserved that for small value of wave number the graph ofphase velocity shows opposite behavior wrt thickness thatis on increasing the values of thickness of layer the values ofcorresponding phase velocity decreases and it became steadyfor large values of wave number

Figure 5 shows the variation of attenuation coefficientwith respect to the wave number for different value ofthickness of the layer Here also we considered the particularcase of fiber-reinforced transversely isotropic thermoelasticmaterial and analyses of the behavior of curveswith respect toinitial stress It is perceived that on decreasing the value of119867the curve shows consistency for small value of wave numberand on increasing the value of119867 the curve shows uniformityfor large value of 120585

Figure 6 depicts the variations of specific loss verseswave number For FTIWIS and FTIIS the curve shows sharpdecrease in the value of specific loss for the range 0 le 120585 le 1and become parallel for 120585 gt 1 It is observed that graphs

8 Journal of Thermodynamics

0 1 2 3 4 5Wave number

25

20

15

10

05

00

Phas

e velo

city

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

times102

Figure 4 Variation of phase velocity wrt wave number (with andwithout initial stress for different value of119867)

corresponding to FTTIWIS and FTTIIS follow the similarpattern for whole range of 120585 but the values are different inmagnitude

81 Rayleigh Wave Figure 7 shows the variations of phasevelocity of Rayleigh wave verses wave number for FTTIWISFTTIIS FTIWIS and FTIIS The value of phase velocitydecreases for small values of 120585 and become stable for highervalue of 120585 It is analyzed that in all the cases behavior of curvesare almost the same within the range 2 le 120585 le 5 Figure 8shows the variations of attenuation coefficient of Rayleighwave verses wave number for FTTIWIS FTTIIS FTIWISand FTIIS For FTIWIS and FTIIS the graph shows a suddenfall for 120585 = 05 and then increases monotonically up to 120585 = 2and becomes constant for the remaining range of 120585 A largedifference between the behavior of curves for the transverselyisotropic and isotropic case is also perceived

9 Conclusions

The propagation of surface waves in an initially stressedfiber-reinforced transversely isotropic thermoelastic mate-rial under an inviscid liquid layer has been studied Afterdeveloping the formal solutions the secular equation forsurface wave propagation is derived Some special casesof frequency equation are also discussed The attenuationcoefficients and specific loss (liquid layer and half-space)have also been studied and computed for FTTIWIS FTTIISFTIWIS and FTIISThe trend of phase velocity is the same inall these four cases for surface wave but varies for attenuation

0 1 2 3 4 5Wave number

00

05

times10minus1

minus15

minus05

minus10

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

Figure 5 Variation of attenuation coeff wrt wave number (withand without initial stress for different value of119867)

0 1 2 3 4 5

Spec

ific l

oss

40

35

30

25

20

15

10

05

00

minus05

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

times10minus4

Figure 6 Variation of specific loss wrt wave number (with andwithout initial stress)

coefficient From numerical analysis it is shown that thephase speeds and attenuation coefficients for Rayleigh waveare also affected significantly due to the presence of initialstress parameters

Journal of Thermodynamics 9

0 1 2 3 4 5

Phas

e velo

city

Wave number

12

10

80

times103

60

40

20

00

FTTIWISFTTIIS

FTIWISFTIIS

Figure 7 Variation of phase velocity wrt wave number forRayleigh wave (with and without initial stress)

0 1 2 3 4 5

00

minus20

minus40

minus60

minus80

minus10

Wave number

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times105

Figure 8 Variation of attenuation coeff wrt wave number forRayleigh wave (with and without initial stress)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A J M Spencer Deformation of Fibre-reinforced MaterialsUniversity of Oxford Clarendon Va USA 1941

[2] A J Belfield T G Rogers and A JM Spencer ldquoStress in elasticplates reinforced by fibres lying in concentric circlesrdquo Journal oftheMechanics and Physics of Solids vol 31 no 1 pp 25ndash54 1983

[3] A C Pipkin ldquoFinite deformations of ideal fiber-reinforcedcompositesrdquo in Composites Materials G P Sendeckyj Ed vol2 of Mechanics of Composite Materials pp 251ndash308 AcademicPress New York NY USA 1973

[4] P R Sengupta and S Nath ldquoSurface waves in fibre-reinforcedanisotropic elastic mediardquo Indian Academy of Sciences SadhanaProceedings vol 26 pp 363ndash370 2001

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

[6] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 198081

[7] A U Erdem ldquoHeat Conduction in fiber-reinforced rigid bod-iesrdquo 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK) 6ndash8Eylul Ankara Turkey 1995

[8] R Kumar and R R Gupta ldquoStudy of wave motion in ananisotropic fiber-reinforced thermoelastic solidrdquo Journal ofSolid Mechanics vol 2 no 1 pp 91ndash100 2010

[9] P Chadwick and L T C Seet ldquoWave propagation in atransversely isotropic heat-conducting elastic materialrdquo Math-ematika vol 17 pp 255ndash274 1970

[10] H Singh and J N Sharma ldquoGeneralized thermoelastic waves intransversely isotropicmediardquo Journal of the Acoustical Society ofAmerica vol 77 pp 1046ndash1053 1985

[11] B Singh ldquoWave propagation in an anisotropic generalizedthermoelastic solidrdquo Indian Journal of Pure and Applied Mathe-matics vol 34 no 10 pp 1479ndash1485 2003

[12] P Chadwick Progress in Solid Mechanics vol 1 North-HollandAmsterdam The Netherlands 1960 edited by R Hill I NSneddon

[13] P Chadwick and DWWindle ldquoPropagation of Rayleigh wavesalong isothermal and insulated boundariesrdquo Proceedings of theRoyal Society of America vol 280 pp 47ndash71 1964

[14] J N Sharma and H Singh ldquoThermoelastic surface waves ina transversely isotropic half space with thermal relaxationsrdquoIndian Journal of Pure and Applied Mathematics vol 16 no 10pp 1202ndash1219 1985

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

[16] J Wang and P Slattery ldquoThermoelasticity without energydissipation for initially stressed bodiesrdquo International Journal ofMathematics and Mathematical Sciences vol 31 no 6 pp 329ndash337 2002

[17] D Iesan ldquoA theory of prestressed thermoelastic Cosserat con-tinuardquo Journal of Applied Mathematics and Mechanics vol 88no 4 pp 306ndash319 2008

[18] K Ames and B Straughan ldquoContinuous dependence results forinitially prestressed thermoelastic bodiesrdquo International Journalof Engineering Science vol 30 no 1 pp 7ndash13 1992

[19] J Wang R S Dhaliwal and S R Majumdar ldquoSome theoremsin the generalized theory of thermoelasticity for prestressedbodiesrdquo Indian Journal of Pure and Applied Mathematics vol28 no 2 pp 267ndash276 1997

[20] M Marin and C Marinescu ldquoThermoelasticity of initiallystressed bodies asymptotic equipartition of energiesrdquo Interna-tional Journal of Engineering Science vol 36 no 1 pp 73ndash861998

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Journal of Thermodynamics

0 1 2 3 4 5Wave number

25

20

15

10

05

00

Phas

e velo

city

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

times102

Figure 4 Variation of phase velocity wrt wave number (with andwithout initial stress for different value of119867)

corresponding to FTTIWIS and FTTIIS follow the similarpattern for whole range of 120585 but the values are different inmagnitude

81 Rayleigh Wave Figure 7 shows the variations of phasevelocity of Rayleigh wave verses wave number for FTTIWISFTTIIS FTIWIS and FTIIS The value of phase velocitydecreases for small values of 120585 and become stable for highervalue of 120585 It is analyzed that in all the cases behavior of curvesare almost the same within the range 2 le 120585 le 5 Figure 8shows the variations of attenuation coefficient of Rayleighwave verses wave number for FTTIWIS FTTIIS FTIWISand FTIIS For FTIWIS and FTIIS the graph shows a suddenfall for 120585 = 05 and then increases monotonically up to 120585 = 2and becomes constant for the remaining range of 120585 A largedifference between the behavior of curves for the transverselyisotropic and isotropic case is also perceived

9 Conclusions

The propagation of surface waves in an initially stressedfiber-reinforced transversely isotropic thermoelastic mate-rial under an inviscid liquid layer has been studied Afterdeveloping the formal solutions the secular equation forsurface wave propagation is derived Some special casesof frequency equation are also discussed The attenuationcoefficients and specific loss (liquid layer and half-space)have also been studied and computed for FTTIWIS FTTIISFTIWIS and FTIISThe trend of phase velocity is the same inall these four cases for surface wave but varies for attenuation

0 1 2 3 4 5Wave number

00

05

times10minus1

minus15

minus05

minus10

minus20

minus25

minus30

Atte

nuat

ion

coeffi

cien

t

H01 wisH01 isH02 wisH02 isH03 wis

H03 isH04 wisH04 isH05 wisH05 is

Figure 5 Variation of attenuation coeff wrt wave number (withand without initial stress for different value of119867)

0 1 2 3 4 5

Spec

ific l

oss

40

35

30

25

20

15

10

05

00

minus05

Wave number

FTTIWISFTTIIS

FTIWISFTIIS

times10minus4

Figure 6 Variation of specific loss wrt wave number (with andwithout initial stress)

coefficient From numerical analysis it is shown that thephase speeds and attenuation coefficients for Rayleigh waveare also affected significantly due to the presence of initialstress parameters

Journal of Thermodynamics 9

0 1 2 3 4 5

Phas

e velo

city

Wave number

12

10

80

times103

60

40

20

00

FTTIWISFTTIIS

FTIWISFTIIS

Figure 7 Variation of phase velocity wrt wave number forRayleigh wave (with and without initial stress)

0 1 2 3 4 5

00

minus20

minus40

minus60

minus80

minus10

Wave number

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times105

Figure 8 Variation of attenuation coeff wrt wave number forRayleigh wave (with and without initial stress)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A J M Spencer Deformation of Fibre-reinforced MaterialsUniversity of Oxford Clarendon Va USA 1941

[2] A J Belfield T G Rogers and A JM Spencer ldquoStress in elasticplates reinforced by fibres lying in concentric circlesrdquo Journal oftheMechanics and Physics of Solids vol 31 no 1 pp 25ndash54 1983

[3] A C Pipkin ldquoFinite deformations of ideal fiber-reinforcedcompositesrdquo in Composites Materials G P Sendeckyj Ed vol2 of Mechanics of Composite Materials pp 251ndash308 AcademicPress New York NY USA 1973

[4] P R Sengupta and S Nath ldquoSurface waves in fibre-reinforcedanisotropic elastic mediardquo Indian Academy of Sciences SadhanaProceedings vol 26 pp 363ndash370 2001

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

[6] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 198081

[7] A U Erdem ldquoHeat Conduction in fiber-reinforced rigid bod-iesrdquo 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK) 6ndash8Eylul Ankara Turkey 1995

[8] R Kumar and R R Gupta ldquoStudy of wave motion in ananisotropic fiber-reinforced thermoelastic solidrdquo Journal ofSolid Mechanics vol 2 no 1 pp 91ndash100 2010

[9] P Chadwick and L T C Seet ldquoWave propagation in atransversely isotropic heat-conducting elastic materialrdquo Math-ematika vol 17 pp 255ndash274 1970

[10] H Singh and J N Sharma ldquoGeneralized thermoelastic waves intransversely isotropicmediardquo Journal of the Acoustical Society ofAmerica vol 77 pp 1046ndash1053 1985

[11] B Singh ldquoWave propagation in an anisotropic generalizedthermoelastic solidrdquo Indian Journal of Pure and Applied Mathe-matics vol 34 no 10 pp 1479ndash1485 2003

[12] P Chadwick Progress in Solid Mechanics vol 1 North-HollandAmsterdam The Netherlands 1960 edited by R Hill I NSneddon

[13] P Chadwick and DWWindle ldquoPropagation of Rayleigh wavesalong isothermal and insulated boundariesrdquo Proceedings of theRoyal Society of America vol 280 pp 47ndash71 1964

[14] J N Sharma and H Singh ldquoThermoelastic surface waves ina transversely isotropic half space with thermal relaxationsrdquoIndian Journal of Pure and Applied Mathematics vol 16 no 10pp 1202ndash1219 1985

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

[16] J Wang and P Slattery ldquoThermoelasticity without energydissipation for initially stressed bodiesrdquo International Journal ofMathematics and Mathematical Sciences vol 31 no 6 pp 329ndash337 2002

[17] D Iesan ldquoA theory of prestressed thermoelastic Cosserat con-tinuardquo Journal of Applied Mathematics and Mechanics vol 88no 4 pp 306ndash319 2008

[18] K Ames and B Straughan ldquoContinuous dependence results forinitially prestressed thermoelastic bodiesrdquo International Journalof Engineering Science vol 30 no 1 pp 7ndash13 1992

[19] J Wang R S Dhaliwal and S R Majumdar ldquoSome theoremsin the generalized theory of thermoelasticity for prestressedbodiesrdquo Indian Journal of Pure and Applied Mathematics vol28 no 2 pp 267ndash276 1997

[20] M Marin and C Marinescu ldquoThermoelasticity of initiallystressed bodies asymptotic equipartition of energiesrdquo Interna-tional Journal of Engineering Science vol 36 no 1 pp 73ndash861998

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Thermodynamics 9

0 1 2 3 4 5

Phas

e velo

city

Wave number

12

10

80

times103

60

40

20

00

FTTIWISFTTIIS

FTIWISFTIIS

Figure 7 Variation of phase velocity wrt wave number forRayleigh wave (with and without initial stress)

0 1 2 3 4 5

00

minus20

minus40

minus60

minus80

minus10

Wave number

Atte

nuat

ion

coeffi

cien

t

FTTIWISFTTIIS

FTIWISFTIIS

times105

Figure 8 Variation of attenuation coeff wrt wave number forRayleigh wave (with and without initial stress)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A J M Spencer Deformation of Fibre-reinforced MaterialsUniversity of Oxford Clarendon Va USA 1941

[2] A J Belfield T G Rogers and A JM Spencer ldquoStress in elasticplates reinforced by fibres lying in concentric circlesrdquo Journal oftheMechanics and Physics of Solids vol 31 no 1 pp 25ndash54 1983

[3] A C Pipkin ldquoFinite deformations of ideal fiber-reinforcedcompositesrdquo in Composites Materials G P Sendeckyj Ed vol2 of Mechanics of Composite Materials pp 251ndash308 AcademicPress New York NY USA 1973

[4] P R Sengupta and S Nath ldquoSurface waves in fibre-reinforcedanisotropic elastic mediardquo Indian Academy of Sciences SadhanaProceedings vol 26 pp 363ndash370 2001

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

[6] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 198081

[7] A U Erdem ldquoHeat Conduction in fiber-reinforced rigid bod-iesrdquo 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK) 6ndash8Eylul Ankara Turkey 1995

[8] R Kumar and R R Gupta ldquoStudy of wave motion in ananisotropic fiber-reinforced thermoelastic solidrdquo Journal ofSolid Mechanics vol 2 no 1 pp 91ndash100 2010

[9] P Chadwick and L T C Seet ldquoWave propagation in atransversely isotropic heat-conducting elastic materialrdquo Math-ematika vol 17 pp 255ndash274 1970

[10] H Singh and J N Sharma ldquoGeneralized thermoelastic waves intransversely isotropicmediardquo Journal of the Acoustical Society ofAmerica vol 77 pp 1046ndash1053 1985

[11] B Singh ldquoWave propagation in an anisotropic generalizedthermoelastic solidrdquo Indian Journal of Pure and Applied Mathe-matics vol 34 no 10 pp 1479ndash1485 2003

[12] P Chadwick Progress in Solid Mechanics vol 1 North-HollandAmsterdam The Netherlands 1960 edited by R Hill I NSneddon

[13] P Chadwick and DWWindle ldquoPropagation of Rayleigh wavesalong isothermal and insulated boundariesrdquo Proceedings of theRoyal Society of America vol 280 pp 47ndash71 1964

[14] J N Sharma and H Singh ldquoThermoelastic surface waves ina transversely isotropic half space with thermal relaxationsrdquoIndian Journal of Pure and Applied Mathematics vol 16 no 10pp 1202ndash1219 1985

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

[16] J Wang and P Slattery ldquoThermoelasticity without energydissipation for initially stressed bodiesrdquo International Journal ofMathematics and Mathematical Sciences vol 31 no 6 pp 329ndash337 2002

[17] D Iesan ldquoA theory of prestressed thermoelastic Cosserat con-tinuardquo Journal of Applied Mathematics and Mechanics vol 88no 4 pp 306ndash319 2008

[18] K Ames and B Straughan ldquoContinuous dependence results forinitially prestressed thermoelastic bodiesrdquo International Journalof Engineering Science vol 30 no 1 pp 7ndash13 1992

[19] J Wang R S Dhaliwal and S R Majumdar ldquoSome theoremsin the generalized theory of thermoelasticity for prestressedbodiesrdquo Indian Journal of Pure and Applied Mathematics vol28 no 2 pp 267ndash276 1997

[20] M Marin and C Marinescu ldquoThermoelasticity of initiallystressed bodies asymptotic equipartition of energiesrdquo Interna-tional Journal of Engineering Science vol 36 no 1 pp 73ndash861998

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Journal of Thermodynamics

[21] V V Kalinchuk T I Belyankova Y E Puzanoff and I AZaitseva ldquoSome dynamic properties of the nonhomogeneousthermoelastic prestressed mediardquo The Journal of the AcousticalSociety of America vol 105 no 2 p 1342 1999

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

[23] B Singh ldquoWave propagation in an initially stressed transverselyisotropic thermoelastic solid half-spacerdquo Applied Mathematicsand Computation vol 217 no 2 pp 705ndash715 2010

[24] M I AOthman SM Said andN Sarker ldquoEffect of hydrostaticinitial stress on a fiber-reinforced thermoelastic medium withfractional derivative heat transferrdquo Multidiscipline Modeling inMaterials and Structures vol 9 no 3 pp 410ndash426 2013

[25] AM Abd-Alla S M Abo-Dahab and A Al-Mullise ldquoeffects ofrotation and gravity field on surface waves in fibre-reinforcedthermoela stic media under four theoriesrdquo Journal of AppliedMathematics vol 2013 Article ID 562369 10 pages 2013

[26] M I A Othman and S Y Atwa ldquoEffect of rotation on afiber-reinforced thermo-elastic under Green-Naghdi theoryand influence of gravityrdquo Meccanica vol 49 no 1 pp 23ndash362014

[27] I A Abbas ldquoGeneralized magneto-thermoelastic interactionin a fiber-reinforced anisotropic hollow cylinderrdquo InternationalJournal of Thermophysics vol 33 no 3 pp 567ndash579 2012

[28] I A Abbas ldquoA GN model for thermoelastic interaction inan unbounded fiber-reinforced anisotropic medium with acircular holerdquo Applied Mathematics Letters vol 26 no 2 pp232ndash239 2013

[29] I A Abbas and A Zenkour ldquoTwo-temperature generalizedthermoelastic interaction in an infinite fiber-reinforcedanisotropic plate containing a circular cavity with tworelaxation timesrdquo Journal of Computational and TheoreticalNanoscience vol 11 no 1 pp 1ndash7 2014

[30] I A Abbas and M I A Othman ldquoGeneralized thermoelasticinteraction in a fiber-reinforced anisotropic half-space underhydrostatic initial stressrdquo Journal of Vibration and Control vol18 no 2 pp 175ndash182 2012

[31] W M Ewing W S Jardetzky and F Press Elastic Waves inLayered Media McGraw-Hill New York NY USA 1957

[32] H Kolsky Stress Waves in Solids Clarendon Press Dover NewYork NY USA 1963

[33] M A Ezzat ldquoFundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect con-ductor cylindrical regionrdquo International Journal of EngineeringScience vol 42 no 13-14 pp 1503ndash1519 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of