Research ArticleEffect of Stiffness on Reflection and Transmission ofWaves at an Interface between Heat Conducting ElasticSolid and Micropolar Fluid Media

Rajneesh Kumar1 S C Rajvanshi23 and Mandeep Kaur23

1 Department of Mathematics Kurukshetra University Kurukshetra 136119 India2 Research Centre Punjab Technical University Kapurthala India3 Department of Applied Sciences Gurukul Vidyapeeth Institute of Engineering and Technology Banur Sector 7Patiala District Punjab 140601 India

Correspondence should be addressed to Mandeep Kaur mandeep1125yahoocom

Received 28 March 2014 Accepted 8 July 2014 Published 29 October 2014

Academic Editor Giuseppe Maruccio

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 is concerned with the propagation of waves at an imperfect boundary of heat conducting elastic solidand micropolar fluid media The amplitude ratios of various reflected and transmitted waves are obtained in a closed form due toincidence of longitudinal wave (P-wave) thermalwave (T-wave) and transversewave (SV-wave)The variation of various amplituderatios with angle of incidence is obtained for normal force stiffness transverse force stiffness thermal contact conductance andperfect bonding Numerical results are shown graphically to depict the effect of stiffness and thermal relaxation times on resultingquantities Some particular cases are also deduced in the present investigation

1 Introduction

The fluids in which coupling between the spin of each fluidparticle and the microscopic velocity field is taken intoaccount are termed asmicropolar fluidsThey represent fluidsconsisting of rigid randomly oriented or spherical particlessuspended in a viscous medium where the deformation offluid particles is ignoredThe theory ofmicrofluids was intro-duced by Eringen [1] A microfluid possesses three gyrationvector fields in addition to its classical translatory degrees offreedom As a subclass of these fluids Eringen introduced themicropolar fluids [2] to describe the physical systems whichdo not fall in the realm of viscous fluids In micropolar fluidsthe local fluid elements were allowed to undergo only rigidrotations without stretch These fluids support couple stressthe body couples and asymmetric stress tensor and possess arotational field which is independent of the velocity of fluidA large class of fluids such as anisotropic fluids liquid crystalswith rigidmoleculesmagnetic fluids cloudwith dustmuddyfluids biological fluids and dirty fluids (dusty air snow)

can be modeled more realistically as micropolar fluids Theimportance ofmicropolar fluids in industrial applications hasmotivated many researchers to extend the study in numerousways to explain various physical effects

Ariman et al [3 4] studied microcontinuum fluidmechanics and its applications and Rıha [5] investigated thetheory of heat-conducting micropolar fluid with microtem-perature Eringen and Kafadar [6] discussed polar fieldtheories Brulin [7] studied linear micropolar media Gorla[8] investigated combined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinder and Eringen [9] studied the theoryof thermo-microstretch fluids and bubbly liquid AydemirandVenart [10] investigated flow of a thermomicropolar fluidwith stretch Ciarletta [11] established two sorts of spatialdecay estimates for describing the spatial behaviour of thesolutions for the flow of a heat-conducting micropolar fluidin a semi-infinite cylindrical pipeThe lateral surface is takento be thermal insulated and the adherence of the fluid tothe lateral boundary is assumed A time-dependent velocity

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014 Article ID 312840 16 pageshttpdxdoiorg1011552014312840

2 International Scholarly Research Notices

and angular velocity profile are prescribed together with thetemperature field at the finite end of the pipe Hsia andCheng [12] studied longitudinal plane waves propagation inelastic micropolar porous media Propagation of transversewaves in elastic micropolar porous semispaces is discussedby Hsia et al [13]

Different researchers studied the problems of reflec-tion and transmission of plane waves at an interface ofmicropolar elastic half-spaces (Tomar andGogna [14ndash16] andKumar et al [17 18]) Singh and Tomar [19] discussed the lon-gitudinalwaves at an interface ofmicropolar fluidmicropolarsolid half-spaces Sun et al [20] studied propagation char-acteristics of longitudinal displacement wave in micropolarfluid with micropolar elastic plate Xu et al [21] discussedreflection and transmission characteristics of coupled wavethrough micropolar elastic solid interlayer in micropolarfluid Dang et al [22] investigated propagation characteristicsof coupled wave through micropolar fluid interlayer inmicropolar elastic solid Fu and Wei [23] investigated wavepropagation through the imperfect interface between twomicropolar solids Sharma and Marin [24] studied reflectionand transmission of waves from imperfect boundary betweentwo heat conducting micropolar thermoelastic solids Vish-wakarma et al [25] studied torsional wave propagation in anearthrsquos crustal layer under the influence of imperfect interface

The purpose of the present study is to study the problemof reflection and transmission of plane waves at an imperfectinterface between heat conducting elastic solid and microp-olar fluid media Effects of stiffness and thermal relaxationtimes on the amplitude ratios for incidence of various planewaves for example longitudinal wave (P-wave) thermalwave (T-wave) and transverse wave (SV-wave) are depictednumerically and shown graphically with angle of incidence

2 Basic Equations

The field equations in a homogeneous and isotropic elasticmedium in the context of generalized theories of thermoelas-ticity without body forces and heat sources are given by Lordand Shulman [26] and Green and Lindsay [27] as

(120582 + 2120583) nabla (nabla sdot ) minus 120583 (nabla times (nabla times ))

minus ](1 + 1205911

120597

120597119905)nabla119879 = 120588

1205972

1205971199052

119870lowastΔ119879 = 120588119888lowast (120597119879

120597119905+ 1205910

1205972119879

1205971199052) + ]119879

0(120597

120597119905+ 12057801205910

1205972

1205971199052) (nabla sdot )

(1)

and the constitutive relations are

119905119894119895= 120582119906119903119903120575119894119895+ 120583 (119906

119894119895+ 119906119895119894) minus ](119879 + 120591

1

120597119879

120597119905) 120575119894119895

119894 119895 119903 = 1 2 3

(2)

where ] = (3120582+2120583)120572119879and themeaning of symbols is defined

in the list at the end of the paperThe thermal relaxation times1205910and 1205911satisfy the inequalities 120591

0ge 1205911ge 0 for G-L theory

only and Δ = 120597212059711990921+ 12059721205971199092

3is the Laplacian operator For

Lord Shulman (L-S) theory 1205780= 1 120591

1= 0 and for Green-

Lindsay (G-L) theory 1205780= 0 1205911gt 0

Following Ciarletta [11] the field equations and theconstitutive relations for heat conducting micropolar fluidswithout body forces body couples and heat sources are givenby

1198631V + (120582119891 + 120583119891) nabla (nabla sdot V) + 119870119891 (nabla times Ψ) minus 119887nabla119879119891

minus 1198880nabla120601lowast119891 = 0

1198632Ψ + (120572119891 + 120573119891) nabla (nabla sdot Ψ) + 119870119891 (nabla times V) = 0

(3)

119870lowast1Δ119879119891 minus 119887119879119891

0(nabla sdot V) = 120588119891119886119879119891

0

120597119879119891

120597119905 (4)

120588119891120597120601lowast119891

120597119905= nabla sdot V (5)

where

1198631= (120583119891 + 119870119891) Δ minus 120588119891

120597

120597119905 119863

2= 120574119891Δ minus 119868

120597

120597119905minus 2119870119891

(6)

such that superscript119891 denotes physical quantities andmate-rial constants related to the fluid and the constitutive relationsare

119905119891119894119895= minus119901120575

119894119895+ 120590119891119894119895

119901 = 119887119879119891 + 1198880120601lowast119891

120590119891119894119895= 120582119891120574

119903119903120575119894119895+ (120583119891 + 119870119891) 120574

119894119895+ 120583119891120574

119895119894

119898119891119894119895= 120572119891120592

119903119903120575119894119895+ 120573119891120592

119895119894+ 120574119891120592

119894119895

(7)

where 120574119894119895= V119895119894+ 120576119895119894119903Ψ119903 120592119894119895= Ψ119895119894 119887 = (3120582119891 + 2120583119891 + 119870119891)120572

119879119891

and symbols are defined in the list at the end of the paper

3 Formulation of the Problem

An imperfect interface of a homogeneous isotropic general-ized thermoelastic half-space (medium119872

1) in contact with

heat conducting micropolar fluid half-space (medium 1198722)

is considered The rectangular Cartesian coordinate systemO119909111990921199093having origin on the surface 119909

3= 0 seperating the

twomedia is taken Let us take the 1199091-axis along the interface

between two half-spaces namely 1198721(0 lt 119909

3lt infin) and

1198722(minusinfin lt 119909

3lt 0) in such a way that 119909

3-axis is pointing

vertically downward into the medium1198721 The geometry of

the problem is shown in Figure 1For two dimensional problem in 119909

11199093-plane we take the

displacement vector velocity vector V and microrotationvelocity vector Ψ as

= (1199061(1199091 1199093) 0 119906

3(1199091 1199093))

V = (V1(1199091 1199093) 0 V

3(1199091 1199093)) Ψ = (0 Ψ

2(1199091 1199093) 0)

(8)

International Scholarly Research Notices 3

The following nondimensional quantities are defined

1199091015840119894=120596lowast119909119894

1198881

1199061015840119894=120588120596lowast1198881

]1198790

119906119894 V1015840

119894=1205881198881

]1198790

V119894

12059510158402=

12058811988821

120596lowast]1198790

1205952 (1199051015840 1205911015840

0 12059110158401) = (120596lowast119905 120596lowast120591

0 120596lowast1205911)

(1198791015840 1198791198911015840) = (119879

1198790

119879119891

1198790

) (1199051015840119894119895 1199051198911015840119894119895) =

1

]1198790

(119905119894119895 119905119891119894119895)

1198981198911015840119894119895=120596lowast

1198881]1198790

119898119891119894119895 120601lowast1198911015840 = 120588120601lowast119891

1198701015840119899=1198881

]1198790

119870119899 1198701015840

119905=1198881

]1198790

119870119905 1198701015840

120579=1

]1198881

119870120579

(9)

where 120596lowast = 120588119888lowast11988821119870lowast 1198882

1= (120582 + 2120583)120588

The displacement components 1199061 1199063and velocity com-

ponents V1 V3are related to the potential functions 120601 120601119891 and

120595 120595119891 in dimensionless form as

(1199061 V1) = (

120597

1205971199091

(120601 120601119891) minus120597

1205971199093

(120595 120595119891))

(1199063 V3) = (

120597

1205971199093

(120601 120601119891) +120597

1205971199091

(120595 120595119891))

(10)

Using (10) in (1) and (3)ndash(5) and with the aid of (8) and (9)(after suppressing the primes) we obtain

nabla2120601 minus (1 + 1205911

120597

120597119905)119879 minus

1205972120601

1205971199052= 0

nabla2120595 minus 1198861

1205972120595

1205971199052= 0

nabla2119879 = (1 + 1205910

120597

120597119905)120597119879

120597119905+ 1205761(120597

120597119905+ 12057801205910

1205972

1205971199052)nabla2120601

nabla2120601119891 minus 1198871119879119891 minus 119887

2120601lowast119891 minus 119887

3

120597120601119891

120597119905= 0

nabla2120595119891 + 1198874Ψ2minus 1198875

120597120595119891

120597119905= 0

nabla2119879119891 minus 1198879nabla2120601119891 minus 119887

10

120597119879119891

120597119905= 0

nabla2Ψ2minus 1198876nabla2120595119891 minus 119887

7Ψ2minus 1198878

120597Ψ2

120597119905= 0

11988711nabla2120601119891 minus

120597

120597119905120601lowast119891 = 0

(11)

where

1198861=12058811988821

120583 120576

1=

]21198790

119870lowast120596lowast120588 119887

1=

11988712058811988821

(120582119891+2120583119891+119870119891) 120596lowast]

1198872=

119888011988821

(120582119891 + 2120583119891 + 119870119891) 120596lowast]1198790

1198873=

12058811989111988821

(120582119891 + 2120583119891 + 119870119891) 120596lowast 119887

4=

119870119891

120583119891 + 119870119891

1198875=

12058811989111988821

(120583119891 + 119870119891) 120596lowast 119887

6=11987011989111988821

120574119891120596lowast2

1198877= 21198876 119887

8=11986811988821

120574119891120596lowast 119887

9=119887]1198791198910

119870lowast1120588120596lowast

11988710=120588119891119886119879119891011988821

119870lowast1120596lowast

11988711=

]1198790

12058811989111988821

(12)

4 Boundary Conditions

The boundary conditions at the interface 1199093= 0 are defined

as

11990511989133= 119870119899(1205971199063

120597119905minus V3) 119905119891

31= 119870119905(1205971199061

120597119905minus V1)

119870lowast1

120597119879119891

1205971199093

= 119870120579(119879 minus 119879119891) 119905

33= 11990511989133 119905

31= 11990511989131

11989811989132= 0 119870lowast

120597119879

1205971199093

= 119870lowast1

120597119879119891

1205971199093

(13)

where 119870119899 119870119905 and 119870

120579are the normal force stiffness trans-

verse force stiffness and thermal contact conductance coef-ficients of unit layer thickness having dimensions N secm3N secm3 and NmsecK

5 Reflection and Transmission

The longitudinal wave (P-wave) or thermal wave (T-wave) ortransverse wave (SV-wave) propagating through the medium1198721which is designated as the region 119909

3gt 0 is considered

The plane wave is taken to be incident at the plane 1199093=

0 with its direction of propagation with angle 1205790normal

to the surface Each incident wave corresponds to reflectedP-wave T-wave and SV-wave in medium 119872

1and trans-

mitted longitudinal wave (L-wave) thermal wave (T-wave)and transverse longitudinal wave coupled with transversemicrorotational wave (C-I and C-II waves) in medium119872

2as

shown in Figure 1

4 International Scholarly Research Notices

T(S2)

L( )

SV(S03)

T(S02)

P(S01)

1205791

1205792

1205793

1205794

12057901205791

12057921205793

M1

M2

0

Alowast

C-II ( )C-I (S3)

x1-axis

x3-axis

S4

S1

Figure 1 Geometry of the problem

In order to solve (11) we assume the solutions of thesystem of the form

120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ2

= 120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ2 119890120580119896(1199091 sin 120579minus1199093 cos 120579)minus120596119905

(14)

where 119896 is the wave number 120596 is the angular frequency and120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ

2are arbitrary constants

Making use of (14) in (11) we obtain

1198814 + 11986311198812 + 119864

1= 0 (15)

1198814 + 11986321198812 + 119864

2= 0

1198814 + 11986331198812 + 119864

3= 0

(16)

where

1198631= minus

1

12059100

[1 + (1 minus 1205801205911120596) 1205761(120580

120596+ 12057801205910)] minus 1

1198641=1

12059100

1198632=120580120596

1198873

(1 minus120580119887211988711

120596) +

120580120596

11988710

(1 +12058011988711198879

1205961198873

)

1198642=minus1205962

119887311988710

(1 minus120580119887211988711

120596)

1198633=120580120596

1198875

+ 120580120596 [(1 minus 120580119887

411988761205961198875)

(1198878+ (120580120596) 119887

7)]

1198643= minus

1205962

(1198878+ (120580120596) 119887

7) 1198875

1198812 =1205962

1198962 120591

00= (

120580

120596+ 1205910)

(17)

Here 1198811 1198812are the velocities of P-wave and T-wave in

medium 1198721and these are roots of (15) and 119881

3= 1radic1198861

is the velocity of SV-wave in medium 1198721 1198811 1198812 1198813 and

1198814are the velocities of transmitted longitudinal wave (L-

wave) thermal wave (T-wave) and transverse longitudinalwave coupled with microrotational wave (C-I and C-II) inmedium119872

2 These are roots of (16)

In view of (14) the appropriate solutions of (11) formedium119872

1and medium119872

2are taken as follows

Medium1198721is as follows

120601 119879 =2

sum119894=1

1 119891119894 [1198780119894119890120580119896119894(1199091 sin 1205790119894minus1199093 cos 1205790119894)minus120596119894119905 + 119875

119894]

120595 = 119878031198901205801198963(1199091 sin 12057903minus1199093 cos 12057903)minus1205963119905+119878

31198901205801198963(1199091 sin 12057903+1199093 cos 12057903)minus1205963119905

(18)

where

119891119894=

12057611205962119894(120580120596119894+ 12057801205910)

minus11198812119894+ (120580120596

119894+ 1205910) minus 1205801205761120596119894(120580120596119894+ 12057801205910) (120580120596119894+ 1205911)

119875119894= 119878119894119890120580119896119894(1199091 sin 120579119894+1199093 cos 120579119894)minus120596119894119905

(19)

Medium1198722is as follows

120601119891 119879119891 120601lowast119891

=2

sum119894=1

1 119891119894 119892119894 119878119894119890120580119896119894(1199091 sin 120579119894minus1199093 cos 120579119894)minus120596119894119905

(20)

120595119891 Ψ2

=4

sum119895=3

1 119891119895 119878119895119890120580119896119895(1199091 sin 120579119895minus1199093 cos 120579119895)minus120596119895119905

(21)

where

119891119894=

minus11988731198879

11988711198879(120580120596119894) + 120580120596

119894(1 minus 120580119887

211988711120596119894) (1119881

119894

2

minus 11988710(120580120596119894))

119891119895=

(11988761198875(120580120596119894))

minus1119881119895

2

minus 11988771205962119895+ 1198878(120580120596119894)+119887411988761205962119895

119892119894=minus12058011988711120596119894

119881119894

2

(22)

and 1198780119894 11987803

are the amplitudes of incident (P-wave T-wave)and SV-wave respectively 119878

119894and 119878

3are the amplitudes of

reflected (P-wave T-wave) and SV-wave and 119878119894 119878119895are the

amplitudes of transmitted longitudinal wave thermal wave

International Scholarly Research Notices 5

and transverse longitudinal wave coupled with transversemicrorotational wave respectively

Equation (20) represents the relation between (120601119891 and119879119891) and (120601119891 and 120601119891lowast)

Snellrsquos law is given by

sin 1205790

1198810

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205794

1198814

(23)

where

11989611198811= 11989621198812= 11989631198813= 11989611198811= 11989621198812= 11989631198813= 11989641198814= 120596

at 1199093= 0(24)

Making use of (18)ndash(21) in the boundary conditions (13) andwith the help of (2) (7) (9) (10) (23) and (24) we obtaina system of seven nonhomogeneous equations which can bewritten as

7

sum119895=1

119886119894119895119885119895= 119884119894 (119894 = 1 2 3 4 5 6 7) (25)

where the values of 119886119894119895are given in the Appendix

(1) For incident P-wave

119860lowast = 11987801 119878

02= 11987803= 0 119884

1= 11988611

1198842= minus11988621 119884

3= minus11988631 119884

4= minus11988641

1198845= 11988651 119884

6= 0 119884

7= 11988671

(26)

(2) For incident T-wave

119860lowast = 11987802 119878

01= 11987803= 0 119884

1= 11988612

1198842= minus11988622 119884

3= minus11988632 119884

4= minus11988642

1198845= 11988652 119884

6= 0 119884

7= 11988672

(27)

(3) For incident SV-wave

119860lowast = 11987803 119878

01= 11987802= 0 119884

1= minus11988613

1198842= 11988623 119884

3= 11988633= 0 119884

4= minus11988643

1198845= minus11988653 119884

6= 0 119884

7= 11988673= 0

1198851=1198781

119860lowast 119885

2=1198782

119860lowast 119885

3=1198783

119860lowast

1198854=1198781

119860lowast 1198855=1198782

119860lowast 1198856=1198783

119860lowast 1198857=1198784

119860lowast

(28)

where 1198851 1198852 and 119885

3are the amplitude ratios of reflected P-

wave T-wave and SV-wave in medium1198721and 119885

4 1198855 1198856

and 1198857are the amplitude ratios of transmitted longitudinal

wave (L-wave) thermal wave (T-wave) and transverse lon-gitudinal wave coupled with transverse microrotational wave(C-I and C-II waves) in medium119872

2

6 Particular Cases

61 Case I Normal Force Stiffness 119870119899= 0 119870119905rarr infin and

119870120579rarr infin in (25) yield the resulting quantities for normal

force stiffness and lead a system of seven nonhomogeneousequations given by (A1) with the changed values of 119886

119894119895as

1198862119894= minus

1205962

1198810

sin 1205790 119886

23=1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= 120580120596

1198810

sin 1205790 119886

25= 120580120596

1198810

sin 1205790

11988626= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790 119886

27= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 119891119894 119886

33= 0 119886

34= minus1198911

11988635= minus1198912 119886

36= 11988637= 0

(29)

62 Case II Transverse Force Stiffness 119870119905= 0119870119899rarr infin and

119870120579rarr infin in (25) provide the case of transverse force stiffness

giving a systemof seven nonhomogeneous equations given by(A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790

11988613= minus

1205962

1198810

sin 1205790 119886

14= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 120580120596

1198810

sin 1205790

11988617= 120580120596

1198810

sin 1205790 119886

3119894= 119891119894 119886

33= 0

11988634= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(30)

63 Case III Thermal Contact Conductance Taking 119870120579=

0 119870119905rarr infin and 119870

119899rarr infin in (25) corresponds to the

case of thermal contact conductance and yields a system of

6 International Scholarly Research Notices

seven nonhomogeneous equations as given by (A1) with thechanged values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

(31)

64 Case IV Perfect Bonding If we take119870119899rarr infin119870

119905rarr infin

and 119870120579rarr infin in (25) we obtain the case of perfect bonding

and yield a system of seven nonhomogeneous equations asgiven by (A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790 119886

3119894= 119891119894

11988633= 0 119886

34= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(32)

65 Subcases

(i) Ifmicropolar heat conducting fluidmedium is absentwe obtain the amplitude ratios at the free surface ofthermoelastic solid with one relaxation time and tworelaxation timesThese results are similar to those obtained by A NSinha and S B Sinha [28] for one relaxation time (L-S theory) and Sinha andElsibai [29] for two relaxationtimes (G-L theory)

(ii) In the absence of upper medium 1198722 we obtain the

amplitude ratios at the free surface of thermoelasticsolid for CT-theory (120578

0= 1205910= 1205911= 0)

These results are similar to those obtained by Deresiewicz[30] for CT-theory

7 Special Cases

(i) If 1205780= 1 120591

1= 0 in (25) (A1) (29) (30) and (31)

then we obtain the corresponding amplitude ratios atan interface of thermoelastic solid with one relaxationtime and heat conducting micropolar fluid half-spacefor normal force stiffness transverse force stiffnessand thermal contact conductance

(ii) If 1205780= 0 120591

1gt 0 in (25) (A1) (29) (30) and

(31) then we obtain the corresponding amplituderatios at an interface of thermoelastic solid withtwo relaxation times and heat conducting micropolarfluid half-space for normal force stiffness transverseforce stiffness and thermal contact conductance

8 Numerical Results and Discussion

The following values of relevant parameters for both the half-spaces for numerical computations are taken

Following Singh and Tomar [19] the values of elasticconstants for medium119872

1are taken as

120582 = 0209730 times 1010Nmminus2 120583 = 091822 times 109Nmminus2

120588 = 00034 times 103 Kgmminus3(33)

International Scholarly Research Notices 7

and thermal parameters are taken from Dhaliwal and Singh[31]

] = 0268 times 107Nmminus2 Kminus1

119888lowast = 104 times 103NmKgminus1 Kminus1

119870lowast = 17 times 102N secminus1 Kminus1 1198790= 0298K

1205910= 0613 times 10minus12 sec 120591

1= 0813 times 10minus12 sec

120596 = 1

(34)

Following Singh and Tomar [19] the values of micropolarconstants for medium119872

2are taken as

120582119891 = 15 times 108N secmminus2 120583119891 = 003 times 108N secmminus2

119870119891 = 0000223 times 108N secmminus2 120574119891 = 00000222N sec

120588119891 = 08 times 103 Kgmminus3 119868 = 000400 times 10minus16N sec2mminus2(35)

Thermal parameters for the medium1198722are taken as of

comparable magnitude

1198791198910= 0196K 119870lowast

1= 089 times 102N secminus1 Kminus1

1198880= 0005 times 1011N sec2mminus6

119886 = 15 times 105m2 secminus2 Kminus2 119887 = 16 times 105Nmminus2 Kminus1(36)

The values of amplitude ratios have been computed atdifferent angles of incidence

In Figures 2ndash22 for L-S theory we represent the solid linefor stiffness (ST1) small dashes line for normal force stiffness(NS1)mediumdashes line for transverse force stiffness (TS1)and dash dot dash line for thermal contact conductance(TCS1) For G-L theory we represent the dash double dotdash line for stiffness (ST2) solid line with center symbolldquoplusrdquo for normal force stiffness (NS2) solid line with centersymbol ldquodiamondrdquo for transverse force stiffness (TS2) andsolid line with center symbol ldquocrossrdquo for thermal contactconductance (TCS2)

81 P-Wave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident P-wave are

shown in Figures 2ndash8Figure 2 shows that the values of |119885

1| for NS1 and

NS2 increase in the whole range The values for ST1 ST2transverse force stiffness and thermal contact conductancedecrease in the whole range except near the grazing inci-dence where the values get increased The values for ST1remain more than the values for TS1 TS2 ST2 TCS1 andTCS2 in the whole range

From Figure 3 it is evident that the values of |1198852| for all

the stiffnesses except ST1 and ST2 decrease in the wholerange The values of |119885

2| for NS1 and NS2 are magnified by

multiplying by 10

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

1

Am

plitu

de ra

tio|Z

1|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 2 Variation of |1198851| with angle of incidence (P-wave)

Figure 4 shows that the values of |1198853| for all the stiffnesses

for L-S theory and G-L theory first increase up to intermedi-ate range and then decrease with the increase in 120579

0and the

values for ST2 are greater than the values for ST1 NS1 NS2TS1 TS2 TCS1 and TCS2 in the whole range The values of|1198852| for NS1 and NS2 are magnified by multiplying by 10FromFigure 5 it is noticed that the values of |119885

4| for all the

stiffnesses start withmaximumvalue at normal incidence andthen decrease to attain minimum value at grazing incidenceThe values of amplitude ratios for ST1 and NS1 are more thanthe values for ST2 and NS2 respectively in the whole rangeThere is slight difference in the values of TCS1 andTCS2 in thewhole rangeThe values of |119885

4| for ST1 ST2 NS1 andNS2 are

magnified by multiplying by 102 and the values for TS1 TS2TCS1 and TCS2 are magnified by multiplying by 10

Figure 6 shows that the values of |1198855| for all the boundary

stiffnesses decrease with increase in 120579 and the values ofamplitude ratio for TS1 are greater than the values for allother boundary stiffnesses in the whole range that shows theeffect of transverse force stiffnessThe values of |119885

5| for all the

stiffnesses are magnified by multiplying by 102From Figure 7 it is evident that the amplitude of |119885

6|

for normal force stiffness increases in the range 0∘ lt 1205790lt

48∘ and then decreases in the further range The values fortransverse force stiffness increase in the range 0∘ lt 120579

0lt 59∘

and for thermal contact conductance increase in the range0∘ lt 120579

0lt 56∘ and then decrease The values of |119885

6| for

all the stiffnesses except TCS1 and TCS2 are magnified by

8 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

Am

plitu

de ra

tio|Z

2|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 3 Variation of |1198852| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

3|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 4 Variation of |1198853| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

4|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 5 Variation of |1198854| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Am

plitu

de ra

tio|Z

5|

Angle of incidence 1205790

Figure 6 Variation of |1198855| with angle of incidence (P-wave)

International Scholarly Research Notices 9

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

Am

plitu

de ra

tio|Z

6|

08

06

1

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 7 Variation of |1198856| with angle of incidence (P-wave)

multiplying by 103 while the values for TCS1 and TCS2 aremagnified by multiplying by 102

Figure 8 depicts that the values of |1198857| for ST1 ST2 TS1

and TS2 increase in the range 0∘ lt 1205790lt 56∘ and decrease in

the remaining range The values for ST2 and TS2 are greaterthan the values for ST1 and TS2 respectively in the wholerange The values for normal force stiffness for G-L theoryremain more than the values for L-S theory The values of|1198857| for ST1 ST2 are magnified by multiplying by 108 the

values for TCS1 TCS2 are magnified by multiplying by 106and the values for NS1 NS2 TS1 and TS2 are magnified bymultiplying by 107

82 T-Wave Incident The values of amplitude ratio |1198851| for

transverse force stiffness and thermal contact conductancedecrease in the whole range with slight increase in the initialrange The values for ST1 and ST2 increase in the range 0∘ lt1205790lt 48∘ and then decreaseThese variations have been shown

in Figure 9 The values for TS1 TS2 TCS1 and TCS2 arereduced by dividing by 10

The values of amplitude ratio |1198852| for NS1 and NS2

increase in the whole range while the values for TS1 TS2ST1 ST2 TCS1 TCS2first decrease and then increase to attainmaximum value at grazing incidence These variations areshown in Figure 10

From Figure 11 it is noticed that the values of |1198853|

for normal force stiffness transverse force stiffness andthermal contact conductance for G-L theory are greaterthan the corresponding values for L-S theory It is seen that

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

7|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 8 Variation of |1198857| with angle of incidence (P-wave)

the values for ST1 are greater than the values for ST2 in therange 0∘ lt 120579

0lt 38∘ and in the further range the values for

ST2 are moreFigure 12 depicts that the values of amplitude ratios |119885

4|

for transverse force stiffness and thermal contact conduc-tance oscillate up to intermediate range and then decreasein the further range to attain minimum value at grazingincidence The values for ST1 ST2 NS1 and NS2 decreasein the whole range The values of |119885

4| for ST1 and ST2 are

magnified bymultiplying by a factor of 102 andNS1 NS2 TS1TS2 TCS1 and TCS2 are magnified by a factor of 10

Figure 13 shows that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is different The values of |1198855| for ST1

and ST2 are magnified by multiplying by a factor of 102 andNS1 NS2 TS1 TS2 TCS1 and TCS2 aremagnified by a factorof 10

From Figure 14 it is evident that the values of |1198856| for

all the boundary stiffnesses increase to attain maximumvalue and then decrease up to grazing incidence The valuesfor NS1 and TS1 are greater than the values for NS2 andTS2 respectively that reveals the thermal relaxation timeeffect The values of |119885

5| for ST1 and ST2 are magnified by

multiplying by a factor of 103 and NS1 NS2 TS1 TS2 TCS1and TCS2 are magnified by a factor of 102

Figure 15 depicts that the values of |1198857| for ST1 and ST2

increase in the interval 0∘ lt 1205790lt 39∘ and then decrease in

the further range It is noticed that there is slight differencein the values of amplitude ratio for L-S and G-L theory

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Scholarly Research Notices

and angular velocity profile are prescribed together with thetemperature field at the finite end of the pipe Hsia andCheng [12] studied longitudinal plane waves propagation inelastic micropolar porous media Propagation of transversewaves in elastic micropolar porous semispaces is discussedby Hsia et al [13]

Different researchers studied the problems of reflec-tion and transmission of plane waves at an interface ofmicropolar elastic half-spaces (Tomar andGogna [14ndash16] andKumar et al [17 18]) Singh and Tomar [19] discussed the lon-gitudinalwaves at an interface ofmicropolar fluidmicropolarsolid half-spaces Sun et al [20] studied propagation char-acteristics of longitudinal displacement wave in micropolarfluid with micropolar elastic plate Xu et al [21] discussedreflection and transmission characteristics of coupled wavethrough micropolar elastic solid interlayer in micropolarfluid Dang et al [22] investigated propagation characteristicsof coupled wave through micropolar fluid interlayer inmicropolar elastic solid Fu and Wei [23] investigated wavepropagation through the imperfect interface between twomicropolar solids Sharma and Marin [24] studied reflectionand transmission of waves from imperfect boundary betweentwo heat conducting micropolar thermoelastic solids Vish-wakarma et al [25] studied torsional wave propagation in anearthrsquos crustal layer under the influence of imperfect interface

The purpose of the present study is to study the problemof reflection and transmission of plane waves at an imperfectinterface between heat conducting elastic solid and microp-olar fluid media Effects of stiffness and thermal relaxationtimes on the amplitude ratios for incidence of various planewaves for example longitudinal wave (P-wave) thermalwave (T-wave) and transverse wave (SV-wave) are depictednumerically and shown graphically with angle of incidence

2 Basic Equations

The field equations in a homogeneous and isotropic elasticmedium in the context of generalized theories of thermoelas-ticity without body forces and heat sources are given by Lordand Shulman [26] and Green and Lindsay [27] as

(120582 + 2120583) nabla (nabla sdot ) minus 120583 (nabla times (nabla times ))

minus ](1 + 1205911

120597

120597119905)nabla119879 = 120588

1205972

1205971199052

119870lowastΔ119879 = 120588119888lowast (120597119879

120597119905+ 1205910

1205972119879

1205971199052) + ]119879

0(120597

120597119905+ 12057801205910

1205972

1205971199052) (nabla sdot )

(1)

and the constitutive relations are

119905119894119895= 120582119906119903119903120575119894119895+ 120583 (119906

119894119895+ 119906119895119894) minus ](119879 + 120591

1

120597119879

120597119905) 120575119894119895

119894 119895 119903 = 1 2 3

(2)

where ] = (3120582+2120583)120572119879and themeaning of symbols is defined

in the list at the end of the paperThe thermal relaxation times1205910and 1205911satisfy the inequalities 120591

0ge 1205911ge 0 for G-L theory

only and Δ = 120597212059711990921+ 12059721205971199092

3is the Laplacian operator For

Lord Shulman (L-S) theory 1205780= 1 120591

1= 0 and for Green-

Lindsay (G-L) theory 1205780= 0 1205911gt 0

Following Ciarletta [11] the field equations and theconstitutive relations for heat conducting micropolar fluidswithout body forces body couples and heat sources are givenby

1198631V + (120582119891 + 120583119891) nabla (nabla sdot V) + 119870119891 (nabla times Ψ) minus 119887nabla119879119891

minus 1198880nabla120601lowast119891 = 0

1198632Ψ + (120572119891 + 120573119891) nabla (nabla sdot Ψ) + 119870119891 (nabla times V) = 0

(3)

119870lowast1Δ119879119891 minus 119887119879119891

0(nabla sdot V) = 120588119891119886119879119891

0

120597119879119891

120597119905 (4)

120588119891120597120601lowast119891

120597119905= nabla sdot V (5)

where

1198631= (120583119891 + 119870119891) Δ minus 120588119891

120597

120597119905 119863

2= 120574119891Δ minus 119868

120597

120597119905minus 2119870119891

(6)

such that superscript119891 denotes physical quantities andmate-rial constants related to the fluid and the constitutive relationsare

119905119891119894119895= minus119901120575

119894119895+ 120590119891119894119895

119901 = 119887119879119891 + 1198880120601lowast119891

120590119891119894119895= 120582119891120574

119903119903120575119894119895+ (120583119891 + 119870119891) 120574

119894119895+ 120583119891120574

119895119894

119898119891119894119895= 120572119891120592

119903119903120575119894119895+ 120573119891120592

119895119894+ 120574119891120592

119894119895

(7)

where 120574119894119895= V119895119894+ 120576119895119894119903Ψ119903 120592119894119895= Ψ119895119894 119887 = (3120582119891 + 2120583119891 + 119870119891)120572

119879119891

and symbols are defined in the list at the end of the paper

3 Formulation of the Problem

An imperfect interface of a homogeneous isotropic general-ized thermoelastic half-space (medium119872

1) in contact with

heat conducting micropolar fluid half-space (medium 1198722)

is considered The rectangular Cartesian coordinate systemO119909111990921199093having origin on the surface 119909

3= 0 seperating the

twomedia is taken Let us take the 1199091-axis along the interface

between two half-spaces namely 1198721(0 lt 119909

3lt infin) and

1198722(minusinfin lt 119909

3lt 0) in such a way that 119909

3-axis is pointing

vertically downward into the medium1198721 The geometry of

the problem is shown in Figure 1For two dimensional problem in 119909

11199093-plane we take the

displacement vector velocity vector V and microrotationvelocity vector Ψ as

= (1199061(1199091 1199093) 0 119906

3(1199091 1199093))

V = (V1(1199091 1199093) 0 V

3(1199091 1199093)) Ψ = (0 Ψ

2(1199091 1199093) 0)

(8)

International Scholarly Research Notices 3

The following nondimensional quantities are defined

1199091015840119894=120596lowast119909119894

1198881

1199061015840119894=120588120596lowast1198881

]1198790

119906119894 V1015840

119894=1205881198881

]1198790

V119894

12059510158402=

12058811988821

120596lowast]1198790

1205952 (1199051015840 1205911015840

0 12059110158401) = (120596lowast119905 120596lowast120591

0 120596lowast1205911)

(1198791015840 1198791198911015840) = (119879

1198790

119879119891

1198790

) (1199051015840119894119895 1199051198911015840119894119895) =

1

]1198790

(119905119894119895 119905119891119894119895)

1198981198911015840119894119895=120596lowast

1198881]1198790

119898119891119894119895 120601lowast1198911015840 = 120588120601lowast119891

1198701015840119899=1198881

]1198790

119870119899 1198701015840

119905=1198881

]1198790

119870119905 1198701015840

120579=1

]1198881

119870120579

(9)

where 120596lowast = 120588119888lowast11988821119870lowast 1198882

1= (120582 + 2120583)120588

The displacement components 1199061 1199063and velocity com-

ponents V1 V3are related to the potential functions 120601 120601119891 and

120595 120595119891 in dimensionless form as

(1199061 V1) = (

120597

1205971199091

(120601 120601119891) minus120597

1205971199093

(120595 120595119891))

(1199063 V3) = (

120597

1205971199093

(120601 120601119891) +120597

1205971199091

(120595 120595119891))

(10)

Using (10) in (1) and (3)ndash(5) and with the aid of (8) and (9)(after suppressing the primes) we obtain

nabla2120601 minus (1 + 1205911

120597

120597119905)119879 minus

1205972120601

1205971199052= 0

nabla2120595 minus 1198861

1205972120595

1205971199052= 0

nabla2119879 = (1 + 1205910

120597

120597119905)120597119879

120597119905+ 1205761(120597

120597119905+ 12057801205910

1205972

1205971199052)nabla2120601

nabla2120601119891 minus 1198871119879119891 minus 119887

2120601lowast119891 minus 119887

3

120597120601119891

120597119905= 0

nabla2120595119891 + 1198874Ψ2minus 1198875

120597120595119891

120597119905= 0

nabla2119879119891 minus 1198879nabla2120601119891 minus 119887

10

120597119879119891

120597119905= 0

nabla2Ψ2minus 1198876nabla2120595119891 minus 119887

7Ψ2minus 1198878

120597Ψ2

120597119905= 0

11988711nabla2120601119891 minus

120597

120597119905120601lowast119891 = 0

(11)

where

1198861=12058811988821

120583 120576

1=

]21198790

119870lowast120596lowast120588 119887

1=

11988712058811988821

(120582119891+2120583119891+119870119891) 120596lowast]

1198872=

119888011988821

(120582119891 + 2120583119891 + 119870119891) 120596lowast]1198790

1198873=

12058811989111988821

(120582119891 + 2120583119891 + 119870119891) 120596lowast 119887

4=

119870119891

120583119891 + 119870119891

1198875=

12058811989111988821

(120583119891 + 119870119891) 120596lowast 119887

6=11987011989111988821

120574119891120596lowast2

1198877= 21198876 119887

8=11986811988821

120574119891120596lowast 119887

9=119887]1198791198910

119870lowast1120588120596lowast

11988710=120588119891119886119879119891011988821

119870lowast1120596lowast

11988711=

]1198790

12058811989111988821

(12)

4 Boundary Conditions

The boundary conditions at the interface 1199093= 0 are defined

as

11990511989133= 119870119899(1205971199063

120597119905minus V3) 119905119891

31= 119870119905(1205971199061

120597119905minus V1)

119870lowast1

120597119879119891

1205971199093

= 119870120579(119879 minus 119879119891) 119905

33= 11990511989133 119905

31= 11990511989131

11989811989132= 0 119870lowast

120597119879

1205971199093

= 119870lowast1

120597119879119891

1205971199093

(13)

where 119870119899 119870119905 and 119870

120579are the normal force stiffness trans-

verse force stiffness and thermal contact conductance coef-ficients of unit layer thickness having dimensions N secm3N secm3 and NmsecK

5 Reflection and Transmission

The longitudinal wave (P-wave) or thermal wave (T-wave) ortransverse wave (SV-wave) propagating through the medium1198721which is designated as the region 119909

3gt 0 is considered

The plane wave is taken to be incident at the plane 1199093=

0 with its direction of propagation with angle 1205790normal

to the surface Each incident wave corresponds to reflectedP-wave T-wave and SV-wave in medium 119872

1and trans-

mitted longitudinal wave (L-wave) thermal wave (T-wave)and transverse longitudinal wave coupled with transversemicrorotational wave (C-I and C-II waves) in medium119872

2as

shown in Figure 1

4 International Scholarly Research Notices

T(S2)

L( )

SV(S03)

T(S02)

P(S01)

1205791

1205792

1205793

1205794

12057901205791

12057921205793

M1

M2

0

Alowast

C-II ( )C-I (S3)

x1-axis

x3-axis

S4

S1

Figure 1 Geometry of the problem

In order to solve (11) we assume the solutions of thesystem of the form

120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ2

= 120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ2 119890120580119896(1199091 sin 120579minus1199093 cos 120579)minus120596119905

(14)

where 119896 is the wave number 120596 is the angular frequency and120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ

2are arbitrary constants

Making use of (14) in (11) we obtain

1198814 + 11986311198812 + 119864

1= 0 (15)

1198814 + 11986321198812 + 119864

2= 0

1198814 + 11986331198812 + 119864

3= 0

(16)

where

1198631= minus

1

12059100

[1 + (1 minus 1205801205911120596) 1205761(120580

120596+ 12057801205910)] minus 1

1198641=1

12059100

1198632=120580120596

1198873

(1 minus120580119887211988711

120596) +

120580120596

11988710

(1 +12058011988711198879

1205961198873

)

1198642=minus1205962

119887311988710

(1 minus120580119887211988711

120596)

1198633=120580120596

1198875

+ 120580120596 [(1 minus 120580119887

411988761205961198875)

(1198878+ (120580120596) 119887

7)]

1198643= minus

1205962

(1198878+ (120580120596) 119887

7) 1198875

1198812 =1205962

1198962 120591

00= (

120580

120596+ 1205910)

(17)

Here 1198811 1198812are the velocities of P-wave and T-wave in

medium 1198721and these are roots of (15) and 119881

3= 1radic1198861

is the velocity of SV-wave in medium 1198721 1198811 1198812 1198813 and

1198814are the velocities of transmitted longitudinal wave (L-

wave) thermal wave (T-wave) and transverse longitudinalwave coupled with microrotational wave (C-I and C-II) inmedium119872

2 These are roots of (16)

In view of (14) the appropriate solutions of (11) formedium119872

1and medium119872

2are taken as follows

Medium1198721is as follows

120601 119879 =2

sum119894=1

1 119891119894 [1198780119894119890120580119896119894(1199091 sin 1205790119894minus1199093 cos 1205790119894)minus120596119894119905 + 119875

119894]

120595 = 119878031198901205801198963(1199091 sin 12057903minus1199093 cos 12057903)minus1205963119905+119878

31198901205801198963(1199091 sin 12057903+1199093 cos 12057903)minus1205963119905

(18)

where

119891119894=

12057611205962119894(120580120596119894+ 12057801205910)

minus11198812119894+ (120580120596

119894+ 1205910) minus 1205801205761120596119894(120580120596119894+ 12057801205910) (120580120596119894+ 1205911)

119875119894= 119878119894119890120580119896119894(1199091 sin 120579119894+1199093 cos 120579119894)minus120596119894119905

(19)

Medium1198722is as follows

120601119891 119879119891 120601lowast119891

=2

sum119894=1

1 119891119894 119892119894 119878119894119890120580119896119894(1199091 sin 120579119894minus1199093 cos 120579119894)minus120596119894119905

(20)

120595119891 Ψ2

=4

sum119895=3

1 119891119895 119878119895119890120580119896119895(1199091 sin 120579119895minus1199093 cos 120579119895)minus120596119895119905

(21)

where

119891119894=

minus11988731198879

11988711198879(120580120596119894) + 120580120596

119894(1 minus 120580119887

211988711120596119894) (1119881

119894

2

minus 11988710(120580120596119894))

119891119895=

(11988761198875(120580120596119894))

minus1119881119895

2

minus 11988771205962119895+ 1198878(120580120596119894)+119887411988761205962119895

119892119894=minus12058011988711120596119894

119881119894

2

(22)

and 1198780119894 11987803

are the amplitudes of incident (P-wave T-wave)and SV-wave respectively 119878

119894and 119878

3are the amplitudes of

reflected (P-wave T-wave) and SV-wave and 119878119894 119878119895are the

amplitudes of transmitted longitudinal wave thermal wave

International Scholarly Research Notices 5

and transverse longitudinal wave coupled with transversemicrorotational wave respectively

Equation (20) represents the relation between (120601119891 and119879119891) and (120601119891 and 120601119891lowast)

Snellrsquos law is given by

sin 1205790

1198810

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205794

1198814

(23)

where

11989611198811= 11989621198812= 11989631198813= 11989611198811= 11989621198812= 11989631198813= 11989641198814= 120596

at 1199093= 0(24)

Making use of (18)ndash(21) in the boundary conditions (13) andwith the help of (2) (7) (9) (10) (23) and (24) we obtaina system of seven nonhomogeneous equations which can bewritten as

7

sum119895=1

119886119894119895119885119895= 119884119894 (119894 = 1 2 3 4 5 6 7) (25)

where the values of 119886119894119895are given in the Appendix

(1) For incident P-wave

119860lowast = 11987801 119878

02= 11987803= 0 119884

1= 11988611

1198842= minus11988621 119884

3= minus11988631 119884

4= minus11988641

1198845= 11988651 119884

6= 0 119884

7= 11988671

(26)

(2) For incident T-wave

119860lowast = 11987802 119878

01= 11987803= 0 119884

1= 11988612

1198842= minus11988622 119884

3= minus11988632 119884

4= minus11988642

1198845= 11988652 119884

6= 0 119884

7= 11988672

(27)

(3) For incident SV-wave

119860lowast = 11987803 119878

01= 11987802= 0 119884

1= minus11988613

1198842= 11988623 119884

3= 11988633= 0 119884

4= minus11988643

1198845= minus11988653 119884

6= 0 119884

7= 11988673= 0

1198851=1198781

119860lowast 119885

2=1198782

119860lowast 119885

3=1198783

119860lowast

1198854=1198781

119860lowast 1198855=1198782

119860lowast 1198856=1198783

119860lowast 1198857=1198784

119860lowast

(28)

where 1198851 1198852 and 119885

3are the amplitude ratios of reflected P-

wave T-wave and SV-wave in medium1198721and 119885

4 1198855 1198856

and 1198857are the amplitude ratios of transmitted longitudinal

wave (L-wave) thermal wave (T-wave) and transverse lon-gitudinal wave coupled with transverse microrotational wave(C-I and C-II waves) in medium119872

2

6 Particular Cases

61 Case I Normal Force Stiffness 119870119899= 0 119870119905rarr infin and

119870120579rarr infin in (25) yield the resulting quantities for normal

force stiffness and lead a system of seven nonhomogeneousequations given by (A1) with the changed values of 119886

119894119895as

1198862119894= minus

1205962

1198810

sin 1205790 119886

23=1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= 120580120596

1198810

sin 1205790 119886

25= 120580120596

1198810

sin 1205790

11988626= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790 119886

27= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 119891119894 119886

33= 0 119886

34= minus1198911

11988635= minus1198912 119886

36= 11988637= 0

(29)

62 Case II Transverse Force Stiffness 119870119905= 0119870119899rarr infin and

119870120579rarr infin in (25) provide the case of transverse force stiffness

giving a systemof seven nonhomogeneous equations given by(A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790

11988613= minus

1205962

1198810

sin 1205790 119886

14= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 120580120596

1198810

sin 1205790

11988617= 120580120596

1198810

sin 1205790 119886

3119894= 119891119894 119886

33= 0

11988634= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(30)

63 Case III Thermal Contact Conductance Taking 119870120579=

0 119870119905rarr infin and 119870

119899rarr infin in (25) corresponds to the

case of thermal contact conductance and yields a system of

6 International Scholarly Research Notices

seven nonhomogeneous equations as given by (A1) with thechanged values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

(31)

64 Case IV Perfect Bonding If we take119870119899rarr infin119870

119905rarr infin

and 119870120579rarr infin in (25) we obtain the case of perfect bonding

and yield a system of seven nonhomogeneous equations asgiven by (A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790 119886

3119894= 119891119894

11988633= 0 119886

34= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(32)

65 Subcases

(i) Ifmicropolar heat conducting fluidmedium is absentwe obtain the amplitude ratios at the free surface ofthermoelastic solid with one relaxation time and tworelaxation timesThese results are similar to those obtained by A NSinha and S B Sinha [28] for one relaxation time (L-S theory) and Sinha andElsibai [29] for two relaxationtimes (G-L theory)

(ii) In the absence of upper medium 1198722 we obtain the

amplitude ratios at the free surface of thermoelasticsolid for CT-theory (120578

0= 1205910= 1205911= 0)

These results are similar to those obtained by Deresiewicz[30] for CT-theory

7 Special Cases

(i) If 1205780= 1 120591

1= 0 in (25) (A1) (29) (30) and (31)

then we obtain the corresponding amplitude ratios atan interface of thermoelastic solid with one relaxationtime and heat conducting micropolar fluid half-spacefor normal force stiffness transverse force stiffnessand thermal contact conductance

(ii) If 1205780= 0 120591

1gt 0 in (25) (A1) (29) (30) and

(31) then we obtain the corresponding amplituderatios at an interface of thermoelastic solid withtwo relaxation times and heat conducting micropolarfluid half-space for normal force stiffness transverseforce stiffness and thermal contact conductance

8 Numerical Results and Discussion

The following values of relevant parameters for both the half-spaces for numerical computations are taken

Following Singh and Tomar [19] the values of elasticconstants for medium119872

1are taken as

120582 = 0209730 times 1010Nmminus2 120583 = 091822 times 109Nmminus2

120588 = 00034 times 103 Kgmminus3(33)

International Scholarly Research Notices 7

and thermal parameters are taken from Dhaliwal and Singh[31]

] = 0268 times 107Nmminus2 Kminus1

119888lowast = 104 times 103NmKgminus1 Kminus1

119870lowast = 17 times 102N secminus1 Kminus1 1198790= 0298K

1205910= 0613 times 10minus12 sec 120591

1= 0813 times 10minus12 sec

120596 = 1

(34)

Following Singh and Tomar [19] the values of micropolarconstants for medium119872

2are taken as

120582119891 = 15 times 108N secmminus2 120583119891 = 003 times 108N secmminus2

119870119891 = 0000223 times 108N secmminus2 120574119891 = 00000222N sec

120588119891 = 08 times 103 Kgmminus3 119868 = 000400 times 10minus16N sec2mminus2(35)

Thermal parameters for the medium1198722are taken as of

comparable magnitude

1198791198910= 0196K 119870lowast

1= 089 times 102N secminus1 Kminus1

1198880= 0005 times 1011N sec2mminus6

119886 = 15 times 105m2 secminus2 Kminus2 119887 = 16 times 105Nmminus2 Kminus1(36)

The values of amplitude ratios have been computed atdifferent angles of incidence

In Figures 2ndash22 for L-S theory we represent the solid linefor stiffness (ST1) small dashes line for normal force stiffness(NS1)mediumdashes line for transverse force stiffness (TS1)and dash dot dash line for thermal contact conductance(TCS1) For G-L theory we represent the dash double dotdash line for stiffness (ST2) solid line with center symbolldquoplusrdquo for normal force stiffness (NS2) solid line with centersymbol ldquodiamondrdquo for transverse force stiffness (TS2) andsolid line with center symbol ldquocrossrdquo for thermal contactconductance (TCS2)

81 P-Wave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident P-wave are

shown in Figures 2ndash8Figure 2 shows that the values of |119885

1| for NS1 and

NS2 increase in the whole range The values for ST1 ST2transverse force stiffness and thermal contact conductancedecrease in the whole range except near the grazing inci-dence where the values get increased The values for ST1remain more than the values for TS1 TS2 ST2 TCS1 andTCS2 in the whole range

From Figure 3 it is evident that the values of |1198852| for all

the stiffnesses except ST1 and ST2 decrease in the wholerange The values of |119885

2| for NS1 and NS2 are magnified by

multiplying by 10

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

1

Am

plitu

de ra

tio|Z

1|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 2 Variation of |1198851| with angle of incidence (P-wave)

Figure 4 shows that the values of |1198853| for all the stiffnesses

for L-S theory and G-L theory first increase up to intermedi-ate range and then decrease with the increase in 120579

0and the

values for ST2 are greater than the values for ST1 NS1 NS2TS1 TS2 TCS1 and TCS2 in the whole range The values of|1198852| for NS1 and NS2 are magnified by multiplying by 10FromFigure 5 it is noticed that the values of |119885

4| for all the

stiffnesses start withmaximumvalue at normal incidence andthen decrease to attain minimum value at grazing incidenceThe values of amplitude ratios for ST1 and NS1 are more thanthe values for ST2 and NS2 respectively in the whole rangeThere is slight difference in the values of TCS1 andTCS2 in thewhole rangeThe values of |119885

4| for ST1 ST2 NS1 andNS2 are

magnified by multiplying by 102 and the values for TS1 TS2TCS1 and TCS2 are magnified by multiplying by 10

Figure 6 shows that the values of |1198855| for all the boundary

stiffnesses decrease with increase in 120579 and the values ofamplitude ratio for TS1 are greater than the values for allother boundary stiffnesses in the whole range that shows theeffect of transverse force stiffnessThe values of |119885

5| for all the

stiffnesses are magnified by multiplying by 102From Figure 7 it is evident that the amplitude of |119885

6|

for normal force stiffness increases in the range 0∘ lt 1205790lt

48∘ and then decreases in the further range The values fortransverse force stiffness increase in the range 0∘ lt 120579

0lt 59∘

and for thermal contact conductance increase in the range0∘ lt 120579

0lt 56∘ and then decrease The values of |119885

6| for

all the stiffnesses except TCS1 and TCS2 are magnified by

8 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

Am

plitu

de ra

tio|Z

2|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 3 Variation of |1198852| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

3|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 4 Variation of |1198853| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

4|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 5 Variation of |1198854| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Am

plitu

de ra

tio|Z

5|

Angle of incidence 1205790

Figure 6 Variation of |1198855| with angle of incidence (P-wave)

International Scholarly Research Notices 9

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

Am

plitu

de ra

tio|Z

6|

08

06

1

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 7 Variation of |1198856| with angle of incidence (P-wave)

multiplying by 103 while the values for TCS1 and TCS2 aremagnified by multiplying by 102

Figure 8 depicts that the values of |1198857| for ST1 ST2 TS1

and TS2 increase in the range 0∘ lt 1205790lt 56∘ and decrease in

the remaining range The values for ST2 and TS2 are greaterthan the values for ST1 and TS2 respectively in the wholerange The values for normal force stiffness for G-L theoryremain more than the values for L-S theory The values of|1198857| for ST1 ST2 are magnified by multiplying by 108 the

values for TCS1 TCS2 are magnified by multiplying by 106and the values for NS1 NS2 TS1 and TS2 are magnified bymultiplying by 107

82 T-Wave Incident The values of amplitude ratio |1198851| for

transverse force stiffness and thermal contact conductancedecrease in the whole range with slight increase in the initialrange The values for ST1 and ST2 increase in the range 0∘ lt1205790lt 48∘ and then decreaseThese variations have been shown

in Figure 9 The values for TS1 TS2 TCS1 and TCS2 arereduced by dividing by 10

The values of amplitude ratio |1198852| for NS1 and NS2

increase in the whole range while the values for TS1 TS2ST1 ST2 TCS1 TCS2first decrease and then increase to attainmaximum value at grazing incidence These variations areshown in Figure 10

From Figure 11 it is noticed that the values of |1198853|

for normal force stiffness transverse force stiffness andthermal contact conductance for G-L theory are greaterthan the corresponding values for L-S theory It is seen that

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

7|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 8 Variation of |1198857| with angle of incidence (P-wave)

the values for ST1 are greater than the values for ST2 in therange 0∘ lt 120579

0lt 38∘ and in the further range the values for

ST2 are moreFigure 12 depicts that the values of amplitude ratios |119885

4|

for transverse force stiffness and thermal contact conduc-tance oscillate up to intermediate range and then decreasein the further range to attain minimum value at grazingincidence The values for ST1 ST2 NS1 and NS2 decreasein the whole range The values of |119885

4| for ST1 and ST2 are

magnified bymultiplying by a factor of 102 andNS1 NS2 TS1TS2 TCS1 and TCS2 are magnified by a factor of 10

Figure 13 shows that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is different The values of |1198855| for ST1

and ST2 are magnified by multiplying by a factor of 102 andNS1 NS2 TS1 TS2 TCS1 and TCS2 aremagnified by a factorof 10

From Figure 14 it is evident that the values of |1198856| for

all the boundary stiffnesses increase to attain maximumvalue and then decrease up to grazing incidence The valuesfor NS1 and TS1 are greater than the values for NS2 andTS2 respectively that reveals the thermal relaxation timeeffect The values of |119885

5| for ST1 and ST2 are magnified by

multiplying by a factor of 103 and NS1 NS2 TS1 TS2 TCS1and TCS2 are magnified by a factor of 102

Figure 15 depicts that the values of |1198857| for ST1 and ST2

increase in the interval 0∘ lt 1205790lt 39∘ and then decrease in

the further range It is noticed that there is slight differencein the values of amplitude ratio for L-S and G-L theory

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 3

The following nondimensional quantities are defined

1199091015840119894=120596lowast119909119894

1198881

1199061015840119894=120588120596lowast1198881

]1198790

119906119894 V1015840

119894=1205881198881

]1198790

V119894

12059510158402=

12058811988821

120596lowast]1198790

1205952 (1199051015840 1205911015840

0 12059110158401) = (120596lowast119905 120596lowast120591

0 120596lowast1205911)

(1198791015840 1198791198911015840) = (119879

1198790

119879119891

1198790

) (1199051015840119894119895 1199051198911015840119894119895) =

1

]1198790

(119905119894119895 119905119891119894119895)

1198981198911015840119894119895=120596lowast

1198881]1198790

119898119891119894119895 120601lowast1198911015840 = 120588120601lowast119891

1198701015840119899=1198881

]1198790

119870119899 1198701015840

119905=1198881

]1198790

119870119905 1198701015840

120579=1

]1198881

119870120579

(9)

where 120596lowast = 120588119888lowast11988821119870lowast 1198882

1= (120582 + 2120583)120588

The displacement components 1199061 1199063and velocity com-

ponents V1 V3are related to the potential functions 120601 120601119891 and

120595 120595119891 in dimensionless form as

(1199061 V1) = (

120597

1205971199091

(120601 120601119891) minus120597

1205971199093

(120595 120595119891))

(1199063 V3) = (

120597

1205971199093

(120601 120601119891) +120597

1205971199091

(120595 120595119891))

(10)

Using (10) in (1) and (3)ndash(5) and with the aid of (8) and (9)(after suppressing the primes) we obtain

nabla2120601 minus (1 + 1205911

120597

120597119905)119879 minus

1205972120601

1205971199052= 0

nabla2120595 minus 1198861

1205972120595

1205971199052= 0

nabla2119879 = (1 + 1205910

120597

120597119905)120597119879

120597119905+ 1205761(120597

120597119905+ 12057801205910

1205972

1205971199052)nabla2120601

nabla2120601119891 minus 1198871119879119891 minus 119887

2120601lowast119891 minus 119887

3

120597120601119891

120597119905= 0

nabla2120595119891 + 1198874Ψ2minus 1198875

120597120595119891

120597119905= 0

nabla2119879119891 minus 1198879nabla2120601119891 minus 119887

10

120597119879119891

120597119905= 0

nabla2Ψ2minus 1198876nabla2120595119891 minus 119887

7Ψ2minus 1198878

120597Ψ2

120597119905= 0

11988711nabla2120601119891 minus

120597

120597119905120601lowast119891 = 0

(11)

where

1198861=12058811988821

120583 120576

1=

]21198790

119870lowast120596lowast120588 119887

1=

11988712058811988821

(120582119891+2120583119891+119870119891) 120596lowast]

1198872=

119888011988821

(120582119891 + 2120583119891 + 119870119891) 120596lowast]1198790

1198873=

12058811989111988821

(120582119891 + 2120583119891 + 119870119891) 120596lowast 119887

4=

119870119891

120583119891 + 119870119891

1198875=

12058811989111988821

(120583119891 + 119870119891) 120596lowast 119887

6=11987011989111988821

120574119891120596lowast2

1198877= 21198876 119887

8=11986811988821

120574119891120596lowast 119887

9=119887]1198791198910

119870lowast1120588120596lowast

11988710=120588119891119886119879119891011988821

119870lowast1120596lowast

11988711=

]1198790

12058811989111988821

(12)

4 Boundary Conditions

The boundary conditions at the interface 1199093= 0 are defined

as

11990511989133= 119870119899(1205971199063

120597119905minus V3) 119905119891

31= 119870119905(1205971199061

120597119905minus V1)

119870lowast1

120597119879119891

1205971199093

= 119870120579(119879 minus 119879119891) 119905

33= 11990511989133 119905

31= 11990511989131

11989811989132= 0 119870lowast

120597119879

1205971199093

= 119870lowast1

120597119879119891

1205971199093

(13)

where 119870119899 119870119905 and 119870

120579are the normal force stiffness trans-

verse force stiffness and thermal contact conductance coef-ficients of unit layer thickness having dimensions N secm3N secm3 and NmsecK

5 Reflection and Transmission

The longitudinal wave (P-wave) or thermal wave (T-wave) ortransverse wave (SV-wave) propagating through the medium1198721which is designated as the region 119909

3gt 0 is considered

The plane wave is taken to be incident at the plane 1199093=

0 with its direction of propagation with angle 1205790normal

to the surface Each incident wave corresponds to reflectedP-wave T-wave and SV-wave in medium 119872

1and trans-

mitted longitudinal wave (L-wave) thermal wave (T-wave)and transverse longitudinal wave coupled with transversemicrorotational wave (C-I and C-II waves) in medium119872

2as

shown in Figure 1

4 International Scholarly Research Notices

T(S2)

L( )

SV(S03)

T(S02)

P(S01)

1205791

1205792

1205793

1205794

12057901205791

12057921205793

M1

M2

0

Alowast

C-II ( )C-I (S3)

x1-axis

x3-axis

S4

S1

Figure 1 Geometry of the problem

In order to solve (11) we assume the solutions of thesystem of the form

120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ2

= 120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ2 119890120580119896(1199091 sin 120579minus1199093 cos 120579)minus120596119905

(14)

where 119896 is the wave number 120596 is the angular frequency and120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ

2are arbitrary constants

Making use of (14) in (11) we obtain

1198814 + 11986311198812 + 119864

1= 0 (15)

1198814 + 11986321198812 + 119864

2= 0

1198814 + 11986331198812 + 119864

3= 0

(16)

where

1198631= minus

1

12059100

[1 + (1 minus 1205801205911120596) 1205761(120580

120596+ 12057801205910)] minus 1

1198641=1

12059100

1198632=120580120596

1198873

(1 minus120580119887211988711

120596) +

120580120596

11988710

(1 +12058011988711198879

1205961198873

)

1198642=minus1205962

119887311988710

(1 minus120580119887211988711

120596)

1198633=120580120596

1198875

+ 120580120596 [(1 minus 120580119887

411988761205961198875)

(1198878+ (120580120596) 119887

7)]

1198643= minus

1205962

(1198878+ (120580120596) 119887

7) 1198875

1198812 =1205962

1198962 120591

00= (

120580

120596+ 1205910)

(17)

Here 1198811 1198812are the velocities of P-wave and T-wave in

medium 1198721and these are roots of (15) and 119881

3= 1radic1198861

is the velocity of SV-wave in medium 1198721 1198811 1198812 1198813 and

1198814are the velocities of transmitted longitudinal wave (L-

wave) thermal wave (T-wave) and transverse longitudinalwave coupled with microrotational wave (C-I and C-II) inmedium119872

2 These are roots of (16)

In view of (14) the appropriate solutions of (11) formedium119872

1and medium119872

2are taken as follows

Medium1198721is as follows

120601 119879 =2

sum119894=1

1 119891119894 [1198780119894119890120580119896119894(1199091 sin 1205790119894minus1199093 cos 1205790119894)minus120596119894119905 + 119875

119894]

120595 = 119878031198901205801198963(1199091 sin 12057903minus1199093 cos 12057903)minus1205963119905+119878

31198901205801198963(1199091 sin 12057903+1199093 cos 12057903)minus1205963119905

(18)

where

119891119894=

12057611205962119894(120580120596119894+ 12057801205910)

minus11198812119894+ (120580120596

119894+ 1205910) minus 1205801205761120596119894(120580120596119894+ 12057801205910) (120580120596119894+ 1205911)

119875119894= 119878119894119890120580119896119894(1199091 sin 120579119894+1199093 cos 120579119894)minus120596119894119905

(19)

Medium1198722is as follows

120601119891 119879119891 120601lowast119891

=2

sum119894=1

1 119891119894 119892119894 119878119894119890120580119896119894(1199091 sin 120579119894minus1199093 cos 120579119894)minus120596119894119905

(20)

120595119891 Ψ2

=4

sum119895=3

1 119891119895 119878119895119890120580119896119895(1199091 sin 120579119895minus1199093 cos 120579119895)minus120596119895119905

(21)

where

119891119894=

minus11988731198879

11988711198879(120580120596119894) + 120580120596

119894(1 minus 120580119887

211988711120596119894) (1119881

119894

2

minus 11988710(120580120596119894))

119891119895=

(11988761198875(120580120596119894))

minus1119881119895

2

minus 11988771205962119895+ 1198878(120580120596119894)+119887411988761205962119895

119892119894=minus12058011988711120596119894

119881119894

2

(22)

and 1198780119894 11987803

are the amplitudes of incident (P-wave T-wave)and SV-wave respectively 119878

119894and 119878

3are the amplitudes of

reflected (P-wave T-wave) and SV-wave and 119878119894 119878119895are the

amplitudes of transmitted longitudinal wave thermal wave

International Scholarly Research Notices 5

and transverse longitudinal wave coupled with transversemicrorotational wave respectively

Equation (20) represents the relation between (120601119891 and119879119891) and (120601119891 and 120601119891lowast)

Snellrsquos law is given by

sin 1205790

1198810

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205794

1198814

(23)

where

11989611198811= 11989621198812= 11989631198813= 11989611198811= 11989621198812= 11989631198813= 11989641198814= 120596

at 1199093= 0(24)

Making use of (18)ndash(21) in the boundary conditions (13) andwith the help of (2) (7) (9) (10) (23) and (24) we obtaina system of seven nonhomogeneous equations which can bewritten as

7

sum119895=1

119886119894119895119885119895= 119884119894 (119894 = 1 2 3 4 5 6 7) (25)

where the values of 119886119894119895are given in the Appendix

(1) For incident P-wave

119860lowast = 11987801 119878

02= 11987803= 0 119884

1= 11988611

1198842= minus11988621 119884

3= minus11988631 119884

4= minus11988641

1198845= 11988651 119884

6= 0 119884

7= 11988671

(26)

(2) For incident T-wave

119860lowast = 11987802 119878

01= 11987803= 0 119884

1= 11988612

1198842= minus11988622 119884

3= minus11988632 119884

4= minus11988642

1198845= 11988652 119884

6= 0 119884

7= 11988672

(27)

(3) For incident SV-wave

119860lowast = 11987803 119878

01= 11987802= 0 119884

1= minus11988613

1198842= 11988623 119884

3= 11988633= 0 119884

4= minus11988643

1198845= minus11988653 119884

6= 0 119884

7= 11988673= 0

1198851=1198781

119860lowast 119885

2=1198782

119860lowast 119885

3=1198783

119860lowast

1198854=1198781

119860lowast 1198855=1198782

119860lowast 1198856=1198783

119860lowast 1198857=1198784

119860lowast

(28)

where 1198851 1198852 and 119885

3are the amplitude ratios of reflected P-

wave T-wave and SV-wave in medium1198721and 119885

4 1198855 1198856

and 1198857are the amplitude ratios of transmitted longitudinal

wave (L-wave) thermal wave (T-wave) and transverse lon-gitudinal wave coupled with transverse microrotational wave(C-I and C-II waves) in medium119872

2

6 Particular Cases

61 Case I Normal Force Stiffness 119870119899= 0 119870119905rarr infin and

119870120579rarr infin in (25) yield the resulting quantities for normal

force stiffness and lead a system of seven nonhomogeneousequations given by (A1) with the changed values of 119886

119894119895as

1198862119894= minus

1205962

1198810

sin 1205790 119886

23=1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= 120580120596

1198810

sin 1205790 119886

25= 120580120596

1198810

sin 1205790

11988626= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790 119886

27= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 119891119894 119886

33= 0 119886

34= minus1198911

11988635= minus1198912 119886

36= 11988637= 0

(29)

62 Case II Transverse Force Stiffness 119870119905= 0119870119899rarr infin and

119870120579rarr infin in (25) provide the case of transverse force stiffness

giving a systemof seven nonhomogeneous equations given by(A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790

11988613= minus

1205962

1198810

sin 1205790 119886

14= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 120580120596

1198810

sin 1205790

11988617= 120580120596

1198810

sin 1205790 119886

3119894= 119891119894 119886

33= 0

11988634= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(30)

63 Case III Thermal Contact Conductance Taking 119870120579=

0 119870119905rarr infin and 119870

119899rarr infin in (25) corresponds to the

case of thermal contact conductance and yields a system of

6 International Scholarly Research Notices

seven nonhomogeneous equations as given by (A1) with thechanged values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

(31)

64 Case IV Perfect Bonding If we take119870119899rarr infin119870

119905rarr infin

and 119870120579rarr infin in (25) we obtain the case of perfect bonding

and yield a system of seven nonhomogeneous equations asgiven by (A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790 119886

3119894= 119891119894

11988633= 0 119886

34= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(32)

65 Subcases

(i) Ifmicropolar heat conducting fluidmedium is absentwe obtain the amplitude ratios at the free surface ofthermoelastic solid with one relaxation time and tworelaxation timesThese results are similar to those obtained by A NSinha and S B Sinha [28] for one relaxation time (L-S theory) and Sinha andElsibai [29] for two relaxationtimes (G-L theory)

(ii) In the absence of upper medium 1198722 we obtain the

amplitude ratios at the free surface of thermoelasticsolid for CT-theory (120578

0= 1205910= 1205911= 0)

These results are similar to those obtained by Deresiewicz[30] for CT-theory

7 Special Cases

(i) If 1205780= 1 120591

1= 0 in (25) (A1) (29) (30) and (31)

then we obtain the corresponding amplitude ratios atan interface of thermoelastic solid with one relaxationtime and heat conducting micropolar fluid half-spacefor normal force stiffness transverse force stiffnessand thermal contact conductance

(ii) If 1205780= 0 120591

1gt 0 in (25) (A1) (29) (30) and

(31) then we obtain the corresponding amplituderatios at an interface of thermoelastic solid withtwo relaxation times and heat conducting micropolarfluid half-space for normal force stiffness transverseforce stiffness and thermal contact conductance

8 Numerical Results and Discussion

The following values of relevant parameters for both the half-spaces for numerical computations are taken

Following Singh and Tomar [19] the values of elasticconstants for medium119872

1are taken as

120582 = 0209730 times 1010Nmminus2 120583 = 091822 times 109Nmminus2

120588 = 00034 times 103 Kgmminus3(33)

International Scholarly Research Notices 7

and thermal parameters are taken from Dhaliwal and Singh[31]

] = 0268 times 107Nmminus2 Kminus1

119888lowast = 104 times 103NmKgminus1 Kminus1

119870lowast = 17 times 102N secminus1 Kminus1 1198790= 0298K

1205910= 0613 times 10minus12 sec 120591

1= 0813 times 10minus12 sec

120596 = 1

(34)

Following Singh and Tomar [19] the values of micropolarconstants for medium119872

2are taken as

120582119891 = 15 times 108N secmminus2 120583119891 = 003 times 108N secmminus2

119870119891 = 0000223 times 108N secmminus2 120574119891 = 00000222N sec

120588119891 = 08 times 103 Kgmminus3 119868 = 000400 times 10minus16N sec2mminus2(35)

Thermal parameters for the medium1198722are taken as of

comparable magnitude

1198791198910= 0196K 119870lowast

1= 089 times 102N secminus1 Kminus1

1198880= 0005 times 1011N sec2mminus6

119886 = 15 times 105m2 secminus2 Kminus2 119887 = 16 times 105Nmminus2 Kminus1(36)

The values of amplitude ratios have been computed atdifferent angles of incidence

In Figures 2ndash22 for L-S theory we represent the solid linefor stiffness (ST1) small dashes line for normal force stiffness(NS1)mediumdashes line for transverse force stiffness (TS1)and dash dot dash line for thermal contact conductance(TCS1) For G-L theory we represent the dash double dotdash line for stiffness (ST2) solid line with center symbolldquoplusrdquo for normal force stiffness (NS2) solid line with centersymbol ldquodiamondrdquo for transverse force stiffness (TS2) andsolid line with center symbol ldquocrossrdquo for thermal contactconductance (TCS2)

81 P-Wave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident P-wave are

shown in Figures 2ndash8Figure 2 shows that the values of |119885

1| for NS1 and

NS2 increase in the whole range The values for ST1 ST2transverse force stiffness and thermal contact conductancedecrease in the whole range except near the grazing inci-dence where the values get increased The values for ST1remain more than the values for TS1 TS2 ST2 TCS1 andTCS2 in the whole range

From Figure 3 it is evident that the values of |1198852| for all

the stiffnesses except ST1 and ST2 decrease in the wholerange The values of |119885

2| for NS1 and NS2 are magnified by

multiplying by 10

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

1

Am

plitu

de ra

tio|Z

1|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 2 Variation of |1198851| with angle of incidence (P-wave)

Figure 4 shows that the values of |1198853| for all the stiffnesses

for L-S theory and G-L theory first increase up to intermedi-ate range and then decrease with the increase in 120579

0and the

values for ST2 are greater than the values for ST1 NS1 NS2TS1 TS2 TCS1 and TCS2 in the whole range The values of|1198852| for NS1 and NS2 are magnified by multiplying by 10FromFigure 5 it is noticed that the values of |119885

4| for all the

stiffnesses start withmaximumvalue at normal incidence andthen decrease to attain minimum value at grazing incidenceThe values of amplitude ratios for ST1 and NS1 are more thanthe values for ST2 and NS2 respectively in the whole rangeThere is slight difference in the values of TCS1 andTCS2 in thewhole rangeThe values of |119885

4| for ST1 ST2 NS1 andNS2 are

magnified by multiplying by 102 and the values for TS1 TS2TCS1 and TCS2 are magnified by multiplying by 10

Figure 6 shows that the values of |1198855| for all the boundary

stiffnesses decrease with increase in 120579 and the values ofamplitude ratio for TS1 are greater than the values for allother boundary stiffnesses in the whole range that shows theeffect of transverse force stiffnessThe values of |119885

5| for all the

stiffnesses are magnified by multiplying by 102From Figure 7 it is evident that the amplitude of |119885

6|

for normal force stiffness increases in the range 0∘ lt 1205790lt

48∘ and then decreases in the further range The values fortransverse force stiffness increase in the range 0∘ lt 120579

0lt 59∘

and for thermal contact conductance increase in the range0∘ lt 120579

0lt 56∘ and then decrease The values of |119885

6| for

all the stiffnesses except TCS1 and TCS2 are magnified by

8 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

Am

plitu

de ra

tio|Z

2|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 3 Variation of |1198852| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

3|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 4 Variation of |1198853| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

4|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 5 Variation of |1198854| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Am

plitu

de ra

tio|Z

5|

Angle of incidence 1205790

Figure 6 Variation of |1198855| with angle of incidence (P-wave)

International Scholarly Research Notices 9

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

Am

plitu

de ra

tio|Z

6|

08

06

1

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 7 Variation of |1198856| with angle of incidence (P-wave)

multiplying by 103 while the values for TCS1 and TCS2 aremagnified by multiplying by 102

Figure 8 depicts that the values of |1198857| for ST1 ST2 TS1

and TS2 increase in the range 0∘ lt 1205790lt 56∘ and decrease in

the remaining range The values for ST2 and TS2 are greaterthan the values for ST1 and TS2 respectively in the wholerange The values for normal force stiffness for G-L theoryremain more than the values for L-S theory The values of|1198857| for ST1 ST2 are magnified by multiplying by 108 the

values for TCS1 TCS2 are magnified by multiplying by 106and the values for NS1 NS2 TS1 and TS2 are magnified bymultiplying by 107

82 T-Wave Incident The values of amplitude ratio |1198851| for

transverse force stiffness and thermal contact conductancedecrease in the whole range with slight increase in the initialrange The values for ST1 and ST2 increase in the range 0∘ lt1205790lt 48∘ and then decreaseThese variations have been shown

in Figure 9 The values for TS1 TS2 TCS1 and TCS2 arereduced by dividing by 10

The values of amplitude ratio |1198852| for NS1 and NS2

increase in the whole range while the values for TS1 TS2ST1 ST2 TCS1 TCS2first decrease and then increase to attainmaximum value at grazing incidence These variations areshown in Figure 10

From Figure 11 it is noticed that the values of |1198853|

for normal force stiffness transverse force stiffness andthermal contact conductance for G-L theory are greaterthan the corresponding values for L-S theory It is seen that

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

7|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 8 Variation of |1198857| with angle of incidence (P-wave)

the values for ST1 are greater than the values for ST2 in therange 0∘ lt 120579

0lt 38∘ and in the further range the values for

ST2 are moreFigure 12 depicts that the values of amplitude ratios |119885

4|

for transverse force stiffness and thermal contact conduc-tance oscillate up to intermediate range and then decreasein the further range to attain minimum value at grazingincidence The values for ST1 ST2 NS1 and NS2 decreasein the whole range The values of |119885

4| for ST1 and ST2 are

magnified bymultiplying by a factor of 102 andNS1 NS2 TS1TS2 TCS1 and TCS2 are magnified by a factor of 10

Figure 13 shows that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is different The values of |1198855| for ST1

and ST2 are magnified by multiplying by a factor of 102 andNS1 NS2 TS1 TS2 TCS1 and TCS2 aremagnified by a factorof 10

From Figure 14 it is evident that the values of |1198856| for

all the boundary stiffnesses increase to attain maximumvalue and then decrease up to grazing incidence The valuesfor NS1 and TS1 are greater than the values for NS2 andTS2 respectively that reveals the thermal relaxation timeeffect The values of |119885

5| for ST1 and ST2 are magnified by

multiplying by a factor of 103 and NS1 NS2 TS1 TS2 TCS1and TCS2 are magnified by a factor of 102

Figure 15 depicts that the values of |1198857| for ST1 and ST2

increase in the interval 0∘ lt 1205790lt 39∘ and then decrease in

the further range It is noticed that there is slight differencein the values of amplitude ratio for L-S and G-L theory

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Scholarly Research Notices

T(S2)

L( )

SV(S03)

T(S02)

P(S01)

1205791

1205792

1205793

1205794

12057901205791

12057921205793

M1

M2

0

Alowast

C-II ( )C-I (S3)

x1-axis

x3-axis

S4

S1

Figure 1 Geometry of the problem

In order to solve (11) we assume the solutions of thesystem of the form

120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ2

= 120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ2 119890120580119896(1199091 sin 120579minus1199093 cos 120579)minus120596119905

(14)

where 119896 is the wave number 120596 is the angular frequency and120601 119879 120595 120601119891 120601lowast119891 119879119891 120595119891 Ψ

2are arbitrary constants

Making use of (14) in (11) we obtain

1198814 + 11986311198812 + 119864

1= 0 (15)

1198814 + 11986321198812 + 119864

2= 0

1198814 + 11986331198812 + 119864

3= 0

(16)

where

1198631= minus

1

12059100

[1 + (1 minus 1205801205911120596) 1205761(120580

120596+ 12057801205910)] minus 1

1198641=1

12059100

1198632=120580120596

1198873

(1 minus120580119887211988711

120596) +

120580120596

11988710

(1 +12058011988711198879

1205961198873

)

1198642=minus1205962

119887311988710

(1 minus120580119887211988711

120596)

1198633=120580120596

1198875

+ 120580120596 [(1 minus 120580119887

411988761205961198875)

(1198878+ (120580120596) 119887

7)]

1198643= minus

1205962

(1198878+ (120580120596) 119887

7) 1198875

1198812 =1205962

1198962 120591

00= (

120580

120596+ 1205910)

(17)

Here 1198811 1198812are the velocities of P-wave and T-wave in

medium 1198721and these are roots of (15) and 119881

3= 1radic1198861

is the velocity of SV-wave in medium 1198721 1198811 1198812 1198813 and

1198814are the velocities of transmitted longitudinal wave (L-

wave) thermal wave (T-wave) and transverse longitudinalwave coupled with microrotational wave (C-I and C-II) inmedium119872

2 These are roots of (16)

In view of (14) the appropriate solutions of (11) formedium119872

1and medium119872

2are taken as follows

Medium1198721is as follows

120601 119879 =2

sum119894=1

1 119891119894 [1198780119894119890120580119896119894(1199091 sin 1205790119894minus1199093 cos 1205790119894)minus120596119894119905 + 119875

119894]

120595 = 119878031198901205801198963(1199091 sin 12057903minus1199093 cos 12057903)minus1205963119905+119878

31198901205801198963(1199091 sin 12057903+1199093 cos 12057903)minus1205963119905

(18)

where

119891119894=

12057611205962119894(120580120596119894+ 12057801205910)

minus11198812119894+ (120580120596

119894+ 1205910) minus 1205801205761120596119894(120580120596119894+ 12057801205910) (120580120596119894+ 1205911)

119875119894= 119878119894119890120580119896119894(1199091 sin 120579119894+1199093 cos 120579119894)minus120596119894119905

(19)

Medium1198722is as follows

120601119891 119879119891 120601lowast119891

=2

sum119894=1

1 119891119894 119892119894 119878119894119890120580119896119894(1199091 sin 120579119894minus1199093 cos 120579119894)minus120596119894119905

(20)

120595119891 Ψ2

=4

sum119895=3

1 119891119895 119878119895119890120580119896119895(1199091 sin 120579119895minus1199093 cos 120579119895)minus120596119895119905

(21)

where

119891119894=

minus11988731198879

11988711198879(120580120596119894) + 120580120596

119894(1 minus 120580119887

211988711120596119894) (1119881

119894

2

minus 11988710(120580120596119894))

119891119895=

(11988761198875(120580120596119894))

minus1119881119895

2

minus 11988771205962119895+ 1198878(120580120596119894)+119887411988761205962119895

119892119894=minus12058011988711120596119894

119881119894

2

(22)

and 1198780119894 11987803

are the amplitudes of incident (P-wave T-wave)and SV-wave respectively 119878

119894and 119878

3are the amplitudes of

reflected (P-wave T-wave) and SV-wave and 119878119894 119878119895are the

amplitudes of transmitted longitudinal wave thermal wave

International Scholarly Research Notices 5

and transverse longitudinal wave coupled with transversemicrorotational wave respectively

Equation (20) represents the relation between (120601119891 and119879119891) and (120601119891 and 120601119891lowast)

Snellrsquos law is given by

sin 1205790

1198810

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205794

1198814

(23)

where

11989611198811= 11989621198812= 11989631198813= 11989611198811= 11989621198812= 11989631198813= 11989641198814= 120596

at 1199093= 0(24)

Making use of (18)ndash(21) in the boundary conditions (13) andwith the help of (2) (7) (9) (10) (23) and (24) we obtaina system of seven nonhomogeneous equations which can bewritten as

7

sum119895=1

119886119894119895119885119895= 119884119894 (119894 = 1 2 3 4 5 6 7) (25)

where the values of 119886119894119895are given in the Appendix

(1) For incident P-wave

119860lowast = 11987801 119878

02= 11987803= 0 119884

1= 11988611

1198842= minus11988621 119884

3= minus11988631 119884

4= minus11988641

1198845= 11988651 119884

6= 0 119884

7= 11988671

(26)

(2) For incident T-wave

119860lowast = 11987802 119878

01= 11987803= 0 119884

1= 11988612

1198842= minus11988622 119884

3= minus11988632 119884

4= minus11988642

1198845= 11988652 119884

6= 0 119884

7= 11988672

(27)

(3) For incident SV-wave

119860lowast = 11987803 119878

01= 11987802= 0 119884

1= minus11988613

1198842= 11988623 119884

3= 11988633= 0 119884

4= minus11988643

1198845= minus11988653 119884

6= 0 119884

7= 11988673= 0

1198851=1198781

119860lowast 119885

2=1198782

119860lowast 119885

3=1198783

119860lowast

1198854=1198781

119860lowast 1198855=1198782

119860lowast 1198856=1198783

119860lowast 1198857=1198784

119860lowast

(28)

where 1198851 1198852 and 119885

3are the amplitude ratios of reflected P-

wave T-wave and SV-wave in medium1198721and 119885

4 1198855 1198856

and 1198857are the amplitude ratios of transmitted longitudinal

wave (L-wave) thermal wave (T-wave) and transverse lon-gitudinal wave coupled with transverse microrotational wave(C-I and C-II waves) in medium119872

2

6 Particular Cases

61 Case I Normal Force Stiffness 119870119899= 0 119870119905rarr infin and

119870120579rarr infin in (25) yield the resulting quantities for normal

force stiffness and lead a system of seven nonhomogeneousequations given by (A1) with the changed values of 119886

119894119895as

1198862119894= minus

1205962

1198810

sin 1205790 119886

23=1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= 120580120596

1198810

sin 1205790 119886

25= 120580120596

1198810

sin 1205790

11988626= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790 119886

27= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 119891119894 119886

33= 0 119886

34= minus1198911

11988635= minus1198912 119886

36= 11988637= 0

(29)

62 Case II Transverse Force Stiffness 119870119905= 0119870119899rarr infin and

119870120579rarr infin in (25) provide the case of transverse force stiffness

giving a systemof seven nonhomogeneous equations given by(A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790

11988613= minus

1205962

1198810

sin 1205790 119886

14= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 120580120596

1198810

sin 1205790

11988617= 120580120596

1198810

sin 1205790 119886

3119894= 119891119894 119886

33= 0

11988634= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(30)

63 Case III Thermal Contact Conductance Taking 119870120579=

0 119870119905rarr infin and 119870

119899rarr infin in (25) corresponds to the

case of thermal contact conductance and yields a system of

6 International Scholarly Research Notices

seven nonhomogeneous equations as given by (A1) with thechanged values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

(31)

64 Case IV Perfect Bonding If we take119870119899rarr infin119870

119905rarr infin

and 119870120579rarr infin in (25) we obtain the case of perfect bonding

and yield a system of seven nonhomogeneous equations asgiven by (A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790 119886

3119894= 119891119894

11988633= 0 119886

34= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(32)

65 Subcases

(i) Ifmicropolar heat conducting fluidmedium is absentwe obtain the amplitude ratios at the free surface ofthermoelastic solid with one relaxation time and tworelaxation timesThese results are similar to those obtained by A NSinha and S B Sinha [28] for one relaxation time (L-S theory) and Sinha andElsibai [29] for two relaxationtimes (G-L theory)

(ii) In the absence of upper medium 1198722 we obtain the

amplitude ratios at the free surface of thermoelasticsolid for CT-theory (120578

0= 1205910= 1205911= 0)

These results are similar to those obtained by Deresiewicz[30] for CT-theory

7 Special Cases

(i) If 1205780= 1 120591

1= 0 in (25) (A1) (29) (30) and (31)

then we obtain the corresponding amplitude ratios atan interface of thermoelastic solid with one relaxationtime and heat conducting micropolar fluid half-spacefor normal force stiffness transverse force stiffnessand thermal contact conductance

(ii) If 1205780= 0 120591

1gt 0 in (25) (A1) (29) (30) and

(31) then we obtain the corresponding amplituderatios at an interface of thermoelastic solid withtwo relaxation times and heat conducting micropolarfluid half-space for normal force stiffness transverseforce stiffness and thermal contact conductance

8 Numerical Results and Discussion

The following values of relevant parameters for both the half-spaces for numerical computations are taken

Following Singh and Tomar [19] the values of elasticconstants for medium119872

1are taken as

120582 = 0209730 times 1010Nmminus2 120583 = 091822 times 109Nmminus2

120588 = 00034 times 103 Kgmminus3(33)

International Scholarly Research Notices 7

and thermal parameters are taken from Dhaliwal and Singh[31]

] = 0268 times 107Nmminus2 Kminus1

119888lowast = 104 times 103NmKgminus1 Kminus1

119870lowast = 17 times 102N secminus1 Kminus1 1198790= 0298K

1205910= 0613 times 10minus12 sec 120591

1= 0813 times 10minus12 sec

120596 = 1

(34)

Following Singh and Tomar [19] the values of micropolarconstants for medium119872

2are taken as

120582119891 = 15 times 108N secmminus2 120583119891 = 003 times 108N secmminus2

119870119891 = 0000223 times 108N secmminus2 120574119891 = 00000222N sec

120588119891 = 08 times 103 Kgmminus3 119868 = 000400 times 10minus16N sec2mminus2(35)

Thermal parameters for the medium1198722are taken as of

comparable magnitude

1198791198910= 0196K 119870lowast

1= 089 times 102N secminus1 Kminus1

1198880= 0005 times 1011N sec2mminus6

119886 = 15 times 105m2 secminus2 Kminus2 119887 = 16 times 105Nmminus2 Kminus1(36)

The values of amplitude ratios have been computed atdifferent angles of incidence

In Figures 2ndash22 for L-S theory we represent the solid linefor stiffness (ST1) small dashes line for normal force stiffness(NS1)mediumdashes line for transverse force stiffness (TS1)and dash dot dash line for thermal contact conductance(TCS1) For G-L theory we represent the dash double dotdash line for stiffness (ST2) solid line with center symbolldquoplusrdquo for normal force stiffness (NS2) solid line with centersymbol ldquodiamondrdquo for transverse force stiffness (TS2) andsolid line with center symbol ldquocrossrdquo for thermal contactconductance (TCS2)

81 P-Wave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident P-wave are

shown in Figures 2ndash8Figure 2 shows that the values of |119885

1| for NS1 and

NS2 increase in the whole range The values for ST1 ST2transverse force stiffness and thermal contact conductancedecrease in the whole range except near the grazing inci-dence where the values get increased The values for ST1remain more than the values for TS1 TS2 ST2 TCS1 andTCS2 in the whole range

From Figure 3 it is evident that the values of |1198852| for all

the stiffnesses except ST1 and ST2 decrease in the wholerange The values of |119885

2| for NS1 and NS2 are magnified by

multiplying by 10

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

1

Am

plitu

de ra

tio|Z

1|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 2 Variation of |1198851| with angle of incidence (P-wave)

Figure 4 shows that the values of |1198853| for all the stiffnesses

for L-S theory and G-L theory first increase up to intermedi-ate range and then decrease with the increase in 120579

0and the

values for ST2 are greater than the values for ST1 NS1 NS2TS1 TS2 TCS1 and TCS2 in the whole range The values of|1198852| for NS1 and NS2 are magnified by multiplying by 10FromFigure 5 it is noticed that the values of |119885

4| for all the

stiffnesses start withmaximumvalue at normal incidence andthen decrease to attain minimum value at grazing incidenceThe values of amplitude ratios for ST1 and NS1 are more thanthe values for ST2 and NS2 respectively in the whole rangeThere is slight difference in the values of TCS1 andTCS2 in thewhole rangeThe values of |119885

4| for ST1 ST2 NS1 andNS2 are

magnified by multiplying by 102 and the values for TS1 TS2TCS1 and TCS2 are magnified by multiplying by 10

Figure 6 shows that the values of |1198855| for all the boundary

stiffnesses decrease with increase in 120579 and the values ofamplitude ratio for TS1 are greater than the values for allother boundary stiffnesses in the whole range that shows theeffect of transverse force stiffnessThe values of |119885

5| for all the

stiffnesses are magnified by multiplying by 102From Figure 7 it is evident that the amplitude of |119885

6|

for normal force stiffness increases in the range 0∘ lt 1205790lt

48∘ and then decreases in the further range The values fortransverse force stiffness increase in the range 0∘ lt 120579

0lt 59∘

and for thermal contact conductance increase in the range0∘ lt 120579

0lt 56∘ and then decrease The values of |119885

6| for

all the stiffnesses except TCS1 and TCS2 are magnified by

8 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

Am

plitu

de ra

tio|Z

2|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 3 Variation of |1198852| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

3|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 4 Variation of |1198853| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

4|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 5 Variation of |1198854| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Am

plitu

de ra

tio|Z

5|

Angle of incidence 1205790

Figure 6 Variation of |1198855| with angle of incidence (P-wave)

International Scholarly Research Notices 9

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

Am

plitu

de ra

tio|Z

6|

08

06

1

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 7 Variation of |1198856| with angle of incidence (P-wave)

multiplying by 103 while the values for TCS1 and TCS2 aremagnified by multiplying by 102

Figure 8 depicts that the values of |1198857| for ST1 ST2 TS1

and TS2 increase in the range 0∘ lt 1205790lt 56∘ and decrease in

the remaining range The values for ST2 and TS2 are greaterthan the values for ST1 and TS2 respectively in the wholerange The values for normal force stiffness for G-L theoryremain more than the values for L-S theory The values of|1198857| for ST1 ST2 are magnified by multiplying by 108 the

values for TCS1 TCS2 are magnified by multiplying by 106and the values for NS1 NS2 TS1 and TS2 are magnified bymultiplying by 107

82 T-Wave Incident The values of amplitude ratio |1198851| for

transverse force stiffness and thermal contact conductancedecrease in the whole range with slight increase in the initialrange The values for ST1 and ST2 increase in the range 0∘ lt1205790lt 48∘ and then decreaseThese variations have been shown

in Figure 9 The values for TS1 TS2 TCS1 and TCS2 arereduced by dividing by 10

The values of amplitude ratio |1198852| for NS1 and NS2

increase in the whole range while the values for TS1 TS2ST1 ST2 TCS1 TCS2first decrease and then increase to attainmaximum value at grazing incidence These variations areshown in Figure 10

From Figure 11 it is noticed that the values of |1198853|

for normal force stiffness transverse force stiffness andthermal contact conductance for G-L theory are greaterthan the corresponding values for L-S theory It is seen that

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

7|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 8 Variation of |1198857| with angle of incidence (P-wave)

the values for ST1 are greater than the values for ST2 in therange 0∘ lt 120579

0lt 38∘ and in the further range the values for

ST2 are moreFigure 12 depicts that the values of amplitude ratios |119885

4|

for transverse force stiffness and thermal contact conduc-tance oscillate up to intermediate range and then decreasein the further range to attain minimum value at grazingincidence The values for ST1 ST2 NS1 and NS2 decreasein the whole range The values of |119885

4| for ST1 and ST2 are

magnified bymultiplying by a factor of 102 andNS1 NS2 TS1TS2 TCS1 and TCS2 are magnified by a factor of 10

Figure 13 shows that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is different The values of |1198855| for ST1

and ST2 are magnified by multiplying by a factor of 102 andNS1 NS2 TS1 TS2 TCS1 and TCS2 aremagnified by a factorof 10

From Figure 14 it is evident that the values of |1198856| for

all the boundary stiffnesses increase to attain maximumvalue and then decrease up to grazing incidence The valuesfor NS1 and TS1 are greater than the values for NS2 andTS2 respectively that reveals the thermal relaxation timeeffect The values of |119885

5| for ST1 and ST2 are magnified by

multiplying by a factor of 103 and NS1 NS2 TS1 TS2 TCS1and TCS2 are magnified by a factor of 102

Figure 15 depicts that the values of |1198857| for ST1 and ST2

increase in the interval 0∘ lt 1205790lt 39∘ and then decrease in

the further range It is noticed that there is slight differencein the values of amplitude ratio for L-S and G-L theory

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 5

and transverse longitudinal wave coupled with transversemicrorotational wave respectively

Equation (20) represents the relation between (120601119891 and119879119891) and (120601119891 and 120601119891lowast)

Snellrsquos law is given by

sin 1205790

1198810

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205791

1198811

=sin 1205792

1198812

=sin 1205793

1198813

=sin 1205794

1198814

(23)

where

11989611198811= 11989621198812= 11989631198813= 11989611198811= 11989621198812= 11989631198813= 11989641198814= 120596

at 1199093= 0(24)

Making use of (18)ndash(21) in the boundary conditions (13) andwith the help of (2) (7) (9) (10) (23) and (24) we obtaina system of seven nonhomogeneous equations which can bewritten as

7

sum119895=1

119886119894119895119885119895= 119884119894 (119894 = 1 2 3 4 5 6 7) (25)

where the values of 119886119894119895are given in the Appendix

(1) For incident P-wave

119860lowast = 11987801 119878

02= 11987803= 0 119884

1= 11988611

1198842= minus11988621 119884

3= minus11988631 119884

4= minus11988641

1198845= 11988651 119884

6= 0 119884

7= 11988671

(26)

(2) For incident T-wave

119860lowast = 11987802 119878

01= 11987803= 0 119884

1= 11988612

1198842= minus11988622 119884

3= minus11988632 119884

4= minus11988642

1198845= 11988652 119884

6= 0 119884

7= 11988672

(27)

(3) For incident SV-wave

119860lowast = 11987803 119878

01= 11987802= 0 119884

1= minus11988613

1198842= 11988623 119884

3= 11988633= 0 119884

4= minus11988643

1198845= minus11988653 119884

6= 0 119884

7= 11988673= 0

1198851=1198781

119860lowast 119885

2=1198782

119860lowast 119885

3=1198783

119860lowast

1198854=1198781

119860lowast 1198855=1198782

119860lowast 1198856=1198783

119860lowast 1198857=1198784

119860lowast

(28)

where 1198851 1198852 and 119885

3are the amplitude ratios of reflected P-

wave T-wave and SV-wave in medium1198721and 119885

4 1198855 1198856

and 1198857are the amplitude ratios of transmitted longitudinal

wave (L-wave) thermal wave (T-wave) and transverse lon-gitudinal wave coupled with transverse microrotational wave(C-I and C-II waves) in medium119872

2

6 Particular Cases

61 Case I Normal Force Stiffness 119870119899= 0 119870119905rarr infin and

119870120579rarr infin in (25) yield the resulting quantities for normal

force stiffness and lead a system of seven nonhomogeneousequations given by (A1) with the changed values of 119886

119894119895as

1198862119894= minus

1205962

1198810

sin 1205790 119886

23=1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= 120580120596

1198810

sin 1205790 119886

25= 120580120596

1198810

sin 1205790

11988626= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790 119886

27= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 119891119894 119886

33= 0 119886

34= minus1198911

11988635= minus1198912 119886

36= 11988637= 0

(29)

62 Case II Transverse Force Stiffness 119870119905= 0119870119899rarr infin and

119870120579rarr infin in (25) provide the case of transverse force stiffness

giving a systemof seven nonhomogeneous equations given by(A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790

11988613= minus

1205962

1198810

sin 1205790 119886

14= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 120580120596

1198810

sin 1205790

11988617= 120580120596

1198810

sin 1205790 119886

3119894= 119891119894 119886

33= 0

11988634= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(30)

63 Case III Thermal Contact Conductance Taking 119870120579=

0 119870119905rarr infin and 119870

119899rarr infin in (25) corresponds to the

case of thermal contact conductance and yields a system of

6 International Scholarly Research Notices

seven nonhomogeneous equations as given by (A1) with thechanged values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

(31)

64 Case IV Perfect Bonding If we take119870119899rarr infin119870

119905rarr infin

and 119870120579rarr infin in (25) we obtain the case of perfect bonding

and yield a system of seven nonhomogeneous equations asgiven by (A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790 119886

3119894= 119891119894

11988633= 0 119886

34= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(32)

65 Subcases

(i) Ifmicropolar heat conducting fluidmedium is absentwe obtain the amplitude ratios at the free surface ofthermoelastic solid with one relaxation time and tworelaxation timesThese results are similar to those obtained by A NSinha and S B Sinha [28] for one relaxation time (L-S theory) and Sinha andElsibai [29] for two relaxationtimes (G-L theory)

(ii) In the absence of upper medium 1198722 we obtain the

amplitude ratios at the free surface of thermoelasticsolid for CT-theory (120578

0= 1205910= 1205911= 0)

These results are similar to those obtained by Deresiewicz[30] for CT-theory

7 Special Cases

(i) If 1205780= 1 120591

1= 0 in (25) (A1) (29) (30) and (31)

then we obtain the corresponding amplitude ratios atan interface of thermoelastic solid with one relaxationtime and heat conducting micropolar fluid half-spacefor normal force stiffness transverse force stiffnessand thermal contact conductance

(ii) If 1205780= 0 120591

1gt 0 in (25) (A1) (29) (30) and

(31) then we obtain the corresponding amplituderatios at an interface of thermoelastic solid withtwo relaxation times and heat conducting micropolarfluid half-space for normal force stiffness transverseforce stiffness and thermal contact conductance

8 Numerical Results and Discussion

The following values of relevant parameters for both the half-spaces for numerical computations are taken

Following Singh and Tomar [19] the values of elasticconstants for medium119872

1are taken as

120582 = 0209730 times 1010Nmminus2 120583 = 091822 times 109Nmminus2

120588 = 00034 times 103 Kgmminus3(33)

International Scholarly Research Notices 7

and thermal parameters are taken from Dhaliwal and Singh[31]

] = 0268 times 107Nmminus2 Kminus1

119888lowast = 104 times 103NmKgminus1 Kminus1

119870lowast = 17 times 102N secminus1 Kminus1 1198790= 0298K

1205910= 0613 times 10minus12 sec 120591

1= 0813 times 10minus12 sec

120596 = 1

(34)

Following Singh and Tomar [19] the values of micropolarconstants for medium119872

2are taken as

120582119891 = 15 times 108N secmminus2 120583119891 = 003 times 108N secmminus2

119870119891 = 0000223 times 108N secmminus2 120574119891 = 00000222N sec

120588119891 = 08 times 103 Kgmminus3 119868 = 000400 times 10minus16N sec2mminus2(35)

Thermal parameters for the medium1198722are taken as of

comparable magnitude

1198791198910= 0196K 119870lowast

1= 089 times 102N secminus1 Kminus1

1198880= 0005 times 1011N sec2mminus6

119886 = 15 times 105m2 secminus2 Kminus2 119887 = 16 times 105Nmminus2 Kminus1(36)

The values of amplitude ratios have been computed atdifferent angles of incidence

In Figures 2ndash22 for L-S theory we represent the solid linefor stiffness (ST1) small dashes line for normal force stiffness(NS1)mediumdashes line for transverse force stiffness (TS1)and dash dot dash line for thermal contact conductance(TCS1) For G-L theory we represent the dash double dotdash line for stiffness (ST2) solid line with center symbolldquoplusrdquo for normal force stiffness (NS2) solid line with centersymbol ldquodiamondrdquo for transverse force stiffness (TS2) andsolid line with center symbol ldquocrossrdquo for thermal contactconductance (TCS2)

81 P-Wave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident P-wave are

shown in Figures 2ndash8Figure 2 shows that the values of |119885

1| for NS1 and

NS2 increase in the whole range The values for ST1 ST2transverse force stiffness and thermal contact conductancedecrease in the whole range except near the grazing inci-dence where the values get increased The values for ST1remain more than the values for TS1 TS2 ST2 TCS1 andTCS2 in the whole range

From Figure 3 it is evident that the values of |1198852| for all

the stiffnesses except ST1 and ST2 decrease in the wholerange The values of |119885

2| for NS1 and NS2 are magnified by

multiplying by 10

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

1

Am

plitu

de ra

tio|Z

1|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 2 Variation of |1198851| with angle of incidence (P-wave)

Figure 4 shows that the values of |1198853| for all the stiffnesses

for L-S theory and G-L theory first increase up to intermedi-ate range and then decrease with the increase in 120579

0and the

values for ST2 are greater than the values for ST1 NS1 NS2TS1 TS2 TCS1 and TCS2 in the whole range The values of|1198852| for NS1 and NS2 are magnified by multiplying by 10FromFigure 5 it is noticed that the values of |119885

4| for all the

stiffnesses start withmaximumvalue at normal incidence andthen decrease to attain minimum value at grazing incidenceThe values of amplitude ratios for ST1 and NS1 are more thanthe values for ST2 and NS2 respectively in the whole rangeThere is slight difference in the values of TCS1 andTCS2 in thewhole rangeThe values of |119885

4| for ST1 ST2 NS1 andNS2 are

magnified by multiplying by 102 and the values for TS1 TS2TCS1 and TCS2 are magnified by multiplying by 10

Figure 6 shows that the values of |1198855| for all the boundary

stiffnesses decrease with increase in 120579 and the values ofamplitude ratio for TS1 are greater than the values for allother boundary stiffnesses in the whole range that shows theeffect of transverse force stiffnessThe values of |119885

5| for all the

stiffnesses are magnified by multiplying by 102From Figure 7 it is evident that the amplitude of |119885

6|

for normal force stiffness increases in the range 0∘ lt 1205790lt

48∘ and then decreases in the further range The values fortransverse force stiffness increase in the range 0∘ lt 120579

0lt 59∘

and for thermal contact conductance increase in the range0∘ lt 120579

0lt 56∘ and then decrease The values of |119885

6| for

all the stiffnesses except TCS1 and TCS2 are magnified by

8 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

Am

plitu

de ra

tio|Z

2|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 3 Variation of |1198852| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

3|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 4 Variation of |1198853| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

4|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 5 Variation of |1198854| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Am

plitu

de ra

tio|Z

5|

Angle of incidence 1205790

Figure 6 Variation of |1198855| with angle of incidence (P-wave)

International Scholarly Research Notices 9

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

Am

plitu

de ra

tio|Z

6|

08

06

1

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 7 Variation of |1198856| with angle of incidence (P-wave)

multiplying by 103 while the values for TCS1 and TCS2 aremagnified by multiplying by 102

Figure 8 depicts that the values of |1198857| for ST1 ST2 TS1

and TS2 increase in the range 0∘ lt 1205790lt 56∘ and decrease in

the remaining range The values for ST2 and TS2 are greaterthan the values for ST1 and TS2 respectively in the wholerange The values for normal force stiffness for G-L theoryremain more than the values for L-S theory The values of|1198857| for ST1 ST2 are magnified by multiplying by 108 the

values for TCS1 TCS2 are magnified by multiplying by 106and the values for NS1 NS2 TS1 and TS2 are magnified bymultiplying by 107

82 T-Wave Incident The values of amplitude ratio |1198851| for

transverse force stiffness and thermal contact conductancedecrease in the whole range with slight increase in the initialrange The values for ST1 and ST2 increase in the range 0∘ lt1205790lt 48∘ and then decreaseThese variations have been shown

in Figure 9 The values for TS1 TS2 TCS1 and TCS2 arereduced by dividing by 10

The values of amplitude ratio |1198852| for NS1 and NS2

increase in the whole range while the values for TS1 TS2ST1 ST2 TCS1 TCS2first decrease and then increase to attainmaximum value at grazing incidence These variations areshown in Figure 10

From Figure 11 it is noticed that the values of |1198853|

for normal force stiffness transverse force stiffness andthermal contact conductance for G-L theory are greaterthan the corresponding values for L-S theory It is seen that

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

7|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 8 Variation of |1198857| with angle of incidence (P-wave)

the values for ST1 are greater than the values for ST2 in therange 0∘ lt 120579

0lt 38∘ and in the further range the values for

ST2 are moreFigure 12 depicts that the values of amplitude ratios |119885

4|

for transverse force stiffness and thermal contact conduc-tance oscillate up to intermediate range and then decreasein the further range to attain minimum value at grazingincidence The values for ST1 ST2 NS1 and NS2 decreasein the whole range The values of |119885

4| for ST1 and ST2 are

magnified bymultiplying by a factor of 102 andNS1 NS2 TS1TS2 TCS1 and TCS2 are magnified by a factor of 10

Figure 13 shows that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is different The values of |1198855| for ST1

and ST2 are magnified by multiplying by a factor of 102 andNS1 NS2 TS1 TS2 TCS1 and TCS2 aremagnified by a factorof 10

From Figure 14 it is evident that the values of |1198856| for

all the boundary stiffnesses increase to attain maximumvalue and then decrease up to grazing incidence The valuesfor NS1 and TS1 are greater than the values for NS2 andTS2 respectively that reveals the thermal relaxation timeeffect The values of |119885

5| for ST1 and ST2 are magnified by

multiplying by a factor of 103 and NS1 NS2 TS1 TS2 TCS1and TCS2 are magnified by a factor of 102

Figure 15 depicts that the values of |1198857| for ST1 and ST2

increase in the interval 0∘ lt 1205790lt 39∘ and then decrease in

the further range It is noticed that there is slight differencein the values of amplitude ratio for L-S and G-L theory

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Stochastic AnalysisInternational Journal of

6 International Scholarly Research Notices

seven nonhomogeneous equations as given by (A1) with thechanged values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790

(31)

64 Case IV Perfect Bonding If we take119870119899rarr infin119870

119905rarr infin

and 119870120579rarr infin in (25) we obtain the case of perfect bonding

and yield a system of seven nonhomogeneous equations asgiven by (A1) with the changed values of 119886

119894119895as

1198861119894= minus

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus

1205962

1198810

sin 1205790

11988614= minus

120580120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

120580120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 11988617= 120580120596

1198810

sin 1205790 119886

2119894= minus

1205962

1198810

sin 1205790

11988623=1205962

1198813

radic1 minus11988123

11988120

sin21205790 119886

24= 120580120596

1198810

sin 1205790

11988625= 120580120596

1198810

sin 1205790 119886

26= 120580120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 120580120596

1198814

radic1 minus11988124

11988120

sin21205790 119886

3119894= 119891119894

11988633= 0 119886

34= minus1198911 119886

35= minus1198912 119886

36= 11988637= 0

(32)

65 Subcases

(i) Ifmicropolar heat conducting fluidmedium is absentwe obtain the amplitude ratios at the free surface ofthermoelastic solid with one relaxation time and tworelaxation timesThese results are similar to those obtained by A NSinha and S B Sinha [28] for one relaxation time (L-S theory) and Sinha andElsibai [29] for two relaxationtimes (G-L theory)

(ii) In the absence of upper medium 1198722 we obtain the

amplitude ratios at the free surface of thermoelasticsolid for CT-theory (120578

0= 1205910= 1205911= 0)

These results are similar to those obtained by Deresiewicz[30] for CT-theory

7 Special Cases

(i) If 1205780= 1 120591

1= 0 in (25) (A1) (29) (30) and (31)

then we obtain the corresponding amplitude ratios atan interface of thermoelastic solid with one relaxationtime and heat conducting micropolar fluid half-spacefor normal force stiffness transverse force stiffnessand thermal contact conductance

(ii) If 1205780= 0 120591

1gt 0 in (25) (A1) (29) (30) and

(31) then we obtain the corresponding amplituderatios at an interface of thermoelastic solid withtwo relaxation times and heat conducting micropolarfluid half-space for normal force stiffness transverseforce stiffness and thermal contact conductance

8 Numerical Results and Discussion

The following values of relevant parameters for both the half-spaces for numerical computations are taken

Following Singh and Tomar [19] the values of elasticconstants for medium119872

1are taken as

120582 = 0209730 times 1010Nmminus2 120583 = 091822 times 109Nmminus2

120588 = 00034 times 103 Kgmminus3(33)

International Scholarly Research Notices 7

and thermal parameters are taken from Dhaliwal and Singh[31]

] = 0268 times 107Nmminus2 Kminus1

119888lowast = 104 times 103NmKgminus1 Kminus1

119870lowast = 17 times 102N secminus1 Kminus1 1198790= 0298K

1205910= 0613 times 10minus12 sec 120591

1= 0813 times 10minus12 sec

120596 = 1

(34)

Following Singh and Tomar [19] the values of micropolarconstants for medium119872

2are taken as

120582119891 = 15 times 108N secmminus2 120583119891 = 003 times 108N secmminus2

119870119891 = 0000223 times 108N secmminus2 120574119891 = 00000222N sec

120588119891 = 08 times 103 Kgmminus3 119868 = 000400 times 10minus16N sec2mminus2(35)

Thermal parameters for the medium1198722are taken as of

comparable magnitude

1198791198910= 0196K 119870lowast

1= 089 times 102N secminus1 Kminus1

1198880= 0005 times 1011N sec2mminus6

119886 = 15 times 105m2 secminus2 Kminus2 119887 = 16 times 105Nmminus2 Kminus1(36)

The values of amplitude ratios have been computed atdifferent angles of incidence

In Figures 2ndash22 for L-S theory we represent the solid linefor stiffness (ST1) small dashes line for normal force stiffness(NS1)mediumdashes line for transverse force stiffness (TS1)and dash dot dash line for thermal contact conductance(TCS1) For G-L theory we represent the dash double dotdash line for stiffness (ST2) solid line with center symbolldquoplusrdquo for normal force stiffness (NS2) solid line with centersymbol ldquodiamondrdquo for transverse force stiffness (TS2) andsolid line with center symbol ldquocrossrdquo for thermal contactconductance (TCS2)

81 P-Wave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident P-wave are

shown in Figures 2ndash8Figure 2 shows that the values of |119885

1| for NS1 and

NS2 increase in the whole range The values for ST1 ST2transverse force stiffness and thermal contact conductancedecrease in the whole range except near the grazing inci-dence where the values get increased The values for ST1remain more than the values for TS1 TS2 ST2 TCS1 andTCS2 in the whole range

From Figure 3 it is evident that the values of |1198852| for all

the stiffnesses except ST1 and ST2 decrease in the wholerange The values of |119885

2| for NS1 and NS2 are magnified by

multiplying by 10

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

1

Am

plitu

de ra

tio|Z

1|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 2 Variation of |1198851| with angle of incidence (P-wave)

Figure 4 shows that the values of |1198853| for all the stiffnesses

for L-S theory and G-L theory first increase up to intermedi-ate range and then decrease with the increase in 120579

0and the

values for ST2 are greater than the values for ST1 NS1 NS2TS1 TS2 TCS1 and TCS2 in the whole range The values of|1198852| for NS1 and NS2 are magnified by multiplying by 10FromFigure 5 it is noticed that the values of |119885

4| for all the

stiffnesses start withmaximumvalue at normal incidence andthen decrease to attain minimum value at grazing incidenceThe values of amplitude ratios for ST1 and NS1 are more thanthe values for ST2 and NS2 respectively in the whole rangeThere is slight difference in the values of TCS1 andTCS2 in thewhole rangeThe values of |119885

4| for ST1 ST2 NS1 andNS2 are

magnified by multiplying by 102 and the values for TS1 TS2TCS1 and TCS2 are magnified by multiplying by 10

Figure 6 shows that the values of |1198855| for all the boundary

stiffnesses decrease with increase in 120579 and the values ofamplitude ratio for TS1 are greater than the values for allother boundary stiffnesses in the whole range that shows theeffect of transverse force stiffnessThe values of |119885

5| for all the

stiffnesses are magnified by multiplying by 102From Figure 7 it is evident that the amplitude of |119885

6|

for normal force stiffness increases in the range 0∘ lt 1205790lt

48∘ and then decreases in the further range The values fortransverse force stiffness increase in the range 0∘ lt 120579

0lt 59∘

and for thermal contact conductance increase in the range0∘ lt 120579

0lt 56∘ and then decrease The values of |119885

6| for

all the stiffnesses except TCS1 and TCS2 are magnified by

8 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

Am

plitu

de ra

tio|Z

2|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 3 Variation of |1198852| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

3|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 4 Variation of |1198853| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

4|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 5 Variation of |1198854| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Am

plitu

de ra

tio|Z

5|

Angle of incidence 1205790

Figure 6 Variation of |1198855| with angle of incidence (P-wave)

International Scholarly Research Notices 9

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

Am

plitu

de ra

tio|Z

6|

08

06

1

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 7 Variation of |1198856| with angle of incidence (P-wave)

multiplying by 103 while the values for TCS1 and TCS2 aremagnified by multiplying by 102

Figure 8 depicts that the values of |1198857| for ST1 ST2 TS1

and TS2 increase in the range 0∘ lt 1205790lt 56∘ and decrease in

the remaining range The values for ST2 and TS2 are greaterthan the values for ST1 and TS2 respectively in the wholerange The values for normal force stiffness for G-L theoryremain more than the values for L-S theory The values of|1198857| for ST1 ST2 are magnified by multiplying by 108 the

values for TCS1 TCS2 are magnified by multiplying by 106and the values for NS1 NS2 TS1 and TS2 are magnified bymultiplying by 107

82 T-Wave Incident The values of amplitude ratio |1198851| for

transverse force stiffness and thermal contact conductancedecrease in the whole range with slight increase in the initialrange The values for ST1 and ST2 increase in the range 0∘ lt1205790lt 48∘ and then decreaseThese variations have been shown

in Figure 9 The values for TS1 TS2 TCS1 and TCS2 arereduced by dividing by 10

The values of amplitude ratio |1198852| for NS1 and NS2

increase in the whole range while the values for TS1 TS2ST1 ST2 TCS1 TCS2first decrease and then increase to attainmaximum value at grazing incidence These variations areshown in Figure 10

From Figure 11 it is noticed that the values of |1198853|

for normal force stiffness transverse force stiffness andthermal contact conductance for G-L theory are greaterthan the corresponding values for L-S theory It is seen that

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

7|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 8 Variation of |1198857| with angle of incidence (P-wave)

the values for ST1 are greater than the values for ST2 in therange 0∘ lt 120579

0lt 38∘ and in the further range the values for

ST2 are moreFigure 12 depicts that the values of amplitude ratios |119885

4|

for transverse force stiffness and thermal contact conduc-tance oscillate up to intermediate range and then decreasein the further range to attain minimum value at grazingincidence The values for ST1 ST2 NS1 and NS2 decreasein the whole range The values of |119885

4| for ST1 and ST2 are

magnified bymultiplying by a factor of 102 andNS1 NS2 TS1TS2 TCS1 and TCS2 are magnified by a factor of 10

Figure 13 shows that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is different The values of |1198855| for ST1

and ST2 are magnified by multiplying by a factor of 102 andNS1 NS2 TS1 TS2 TCS1 and TCS2 aremagnified by a factorof 10

From Figure 14 it is evident that the values of |1198856| for

all the boundary stiffnesses increase to attain maximumvalue and then decrease up to grazing incidence The valuesfor NS1 and TS1 are greater than the values for NS2 andTS2 respectively that reveals the thermal relaxation timeeffect The values of |119885

5| for ST1 and ST2 are magnified by

multiplying by a factor of 103 and NS1 NS2 TS1 TS2 TCS1and TCS2 are magnified by a factor of 102

Figure 15 depicts that the values of |1198857| for ST1 and ST2

increase in the interval 0∘ lt 1205790lt 39∘ and then decrease in

the further range It is noticed that there is slight differencein the values of amplitude ratio for L-S and G-L theory

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 7

and thermal parameters are taken from Dhaliwal and Singh[31]

] = 0268 times 107Nmminus2 Kminus1

119888lowast = 104 times 103NmKgminus1 Kminus1

119870lowast = 17 times 102N secminus1 Kminus1 1198790= 0298K

1205910= 0613 times 10minus12 sec 120591

1= 0813 times 10minus12 sec

120596 = 1

(34)

Following Singh and Tomar [19] the values of micropolarconstants for medium119872

2are taken as

120582119891 = 15 times 108N secmminus2 120583119891 = 003 times 108N secmminus2

119870119891 = 0000223 times 108N secmminus2 120574119891 = 00000222N sec

120588119891 = 08 times 103 Kgmminus3 119868 = 000400 times 10minus16N sec2mminus2(35)

Thermal parameters for the medium1198722are taken as of

comparable magnitude

1198791198910= 0196K 119870lowast

1= 089 times 102N secminus1 Kminus1

1198880= 0005 times 1011N sec2mminus6

119886 = 15 times 105m2 secminus2 Kminus2 119887 = 16 times 105Nmminus2 Kminus1(36)

The values of amplitude ratios have been computed atdifferent angles of incidence

In Figures 2ndash22 for L-S theory we represent the solid linefor stiffness (ST1) small dashes line for normal force stiffness(NS1)mediumdashes line for transverse force stiffness (TS1)and dash dot dash line for thermal contact conductance(TCS1) For G-L theory we represent the dash double dotdash line for stiffness (ST2) solid line with center symbolldquoplusrdquo for normal force stiffness (NS2) solid line with centersymbol ldquodiamondrdquo for transverse force stiffness (TS2) andsolid line with center symbol ldquocrossrdquo for thermal contactconductance (TCS2)

81 P-Wave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident P-wave are

shown in Figures 2ndash8Figure 2 shows that the values of |119885

1| for NS1 and

NS2 increase in the whole range The values for ST1 ST2transverse force stiffness and thermal contact conductancedecrease in the whole range except near the grazing inci-dence where the values get increased The values for ST1remain more than the values for TS1 TS2 ST2 TCS1 andTCS2 in the whole range

From Figure 3 it is evident that the values of |1198852| for all

the stiffnesses except ST1 and ST2 decrease in the wholerange The values of |119885

2| for NS1 and NS2 are magnified by

multiplying by 10

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

1

Am

plitu

de ra

tio|Z

1|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 2 Variation of |1198851| with angle of incidence (P-wave)

Figure 4 shows that the values of |1198853| for all the stiffnesses

for L-S theory and G-L theory first increase up to intermedi-ate range and then decrease with the increase in 120579

0and the

values for ST2 are greater than the values for ST1 NS1 NS2TS1 TS2 TCS1 and TCS2 in the whole range The values of|1198852| for NS1 and NS2 are magnified by multiplying by 10FromFigure 5 it is noticed that the values of |119885

4| for all the

stiffnesses start withmaximumvalue at normal incidence andthen decrease to attain minimum value at grazing incidenceThe values of amplitude ratios for ST1 and NS1 are more thanthe values for ST2 and NS2 respectively in the whole rangeThere is slight difference in the values of TCS1 andTCS2 in thewhole rangeThe values of |119885

4| for ST1 ST2 NS1 andNS2 are

magnified by multiplying by 102 and the values for TS1 TS2TCS1 and TCS2 are magnified by multiplying by 10

Figure 6 shows that the values of |1198855| for all the boundary

stiffnesses decrease with increase in 120579 and the values ofamplitude ratio for TS1 are greater than the values for allother boundary stiffnesses in the whole range that shows theeffect of transverse force stiffnessThe values of |119885

5| for all the

stiffnesses are magnified by multiplying by 102From Figure 7 it is evident that the amplitude of |119885

6|

for normal force stiffness increases in the range 0∘ lt 1205790lt

48∘ and then decreases in the further range The values fortransverse force stiffness increase in the range 0∘ lt 120579

0lt 59∘

and for thermal contact conductance increase in the range0∘ lt 120579

0lt 56∘ and then decrease The values of |119885

6| for

all the stiffnesses except TCS1 and TCS2 are magnified by

8 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

Am

plitu

de ra

tio|Z

2|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 3 Variation of |1198852| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

3|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 4 Variation of |1198853| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

4|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 5 Variation of |1198854| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Am

plitu

de ra

tio|Z

5|

Angle of incidence 1205790

Figure 6 Variation of |1198855| with angle of incidence (P-wave)

International Scholarly Research Notices 9

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

Am

plitu

de ra

tio|Z

6|

08

06

1

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 7 Variation of |1198856| with angle of incidence (P-wave)

multiplying by 103 while the values for TCS1 and TCS2 aremagnified by multiplying by 102

Figure 8 depicts that the values of |1198857| for ST1 ST2 TS1

and TS2 increase in the range 0∘ lt 1205790lt 56∘ and decrease in

the remaining range The values for ST2 and TS2 are greaterthan the values for ST1 and TS2 respectively in the wholerange The values for normal force stiffness for G-L theoryremain more than the values for L-S theory The values of|1198857| for ST1 ST2 are magnified by multiplying by 108 the

values for TCS1 TCS2 are magnified by multiplying by 106and the values for NS1 NS2 TS1 and TS2 are magnified bymultiplying by 107

82 T-Wave Incident The values of amplitude ratio |1198851| for

transverse force stiffness and thermal contact conductancedecrease in the whole range with slight increase in the initialrange The values for ST1 and ST2 increase in the range 0∘ lt1205790lt 48∘ and then decreaseThese variations have been shown

in Figure 9 The values for TS1 TS2 TCS1 and TCS2 arereduced by dividing by 10

The values of amplitude ratio |1198852| for NS1 and NS2

increase in the whole range while the values for TS1 TS2ST1 ST2 TCS1 TCS2first decrease and then increase to attainmaximum value at grazing incidence These variations areshown in Figure 10

From Figure 11 it is noticed that the values of |1198853|

for normal force stiffness transverse force stiffness andthermal contact conductance for G-L theory are greaterthan the corresponding values for L-S theory It is seen that

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

7|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 8 Variation of |1198857| with angle of incidence (P-wave)

the values for ST1 are greater than the values for ST2 in therange 0∘ lt 120579

0lt 38∘ and in the further range the values for

ST2 are moreFigure 12 depicts that the values of amplitude ratios |119885

4|

for transverse force stiffness and thermal contact conduc-tance oscillate up to intermediate range and then decreasein the further range to attain minimum value at grazingincidence The values for ST1 ST2 NS1 and NS2 decreasein the whole range The values of |119885

4| for ST1 and ST2 are

magnified bymultiplying by a factor of 102 andNS1 NS2 TS1TS2 TCS1 and TCS2 are magnified by a factor of 10

Figure 13 shows that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is different The values of |1198855| for ST1

and ST2 are magnified by multiplying by a factor of 102 andNS1 NS2 TS1 TS2 TCS1 and TCS2 aremagnified by a factorof 10

From Figure 14 it is evident that the values of |1198856| for

all the boundary stiffnesses increase to attain maximumvalue and then decrease up to grazing incidence The valuesfor NS1 and TS1 are greater than the values for NS2 andTS2 respectively that reveals the thermal relaxation timeeffect The values of |119885

5| for ST1 and ST2 are magnified by

multiplying by a factor of 103 and NS1 NS2 TS1 TS2 TCS1and TCS2 are magnified by a factor of 102

Figure 15 depicts that the values of |1198857| for ST1 and ST2

increase in the interval 0∘ lt 1205790lt 39∘ and then decrease in

the further range It is noticed that there is slight differencein the values of amplitude ratio for L-S and G-L theory

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

Am

plitu

de ra

tio|Z

2|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 3 Variation of |1198852| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

3|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 4 Variation of |1198853| with angle of incidence (P-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

06

08

Am

plitu

de ra

tio|Z

4|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 5 Variation of |1198854| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Am

plitu

de ra

tio|Z

5|

Angle of incidence 1205790

Figure 6 Variation of |1198855| with angle of incidence (P-wave)

International Scholarly Research Notices 9

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

Am

plitu

de ra

tio|Z

6|

08

06

1

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 7 Variation of |1198856| with angle of incidence (P-wave)

multiplying by 103 while the values for TCS1 and TCS2 aremagnified by multiplying by 102

Figure 8 depicts that the values of |1198857| for ST1 ST2 TS1

and TS2 increase in the range 0∘ lt 1205790lt 56∘ and decrease in

the remaining range The values for ST2 and TS2 are greaterthan the values for ST1 and TS2 respectively in the wholerange The values for normal force stiffness for G-L theoryremain more than the values for L-S theory The values of|1198857| for ST1 ST2 are magnified by multiplying by 108 the

values for TCS1 TCS2 are magnified by multiplying by 106and the values for NS1 NS2 TS1 and TS2 are magnified bymultiplying by 107

82 T-Wave Incident The values of amplitude ratio |1198851| for

transverse force stiffness and thermal contact conductancedecrease in the whole range with slight increase in the initialrange The values for ST1 and ST2 increase in the range 0∘ lt1205790lt 48∘ and then decreaseThese variations have been shown

in Figure 9 The values for TS1 TS2 TCS1 and TCS2 arereduced by dividing by 10

The values of amplitude ratio |1198852| for NS1 and NS2

increase in the whole range while the values for TS1 TS2ST1 ST2 TCS1 TCS2first decrease and then increase to attainmaximum value at grazing incidence These variations areshown in Figure 10

From Figure 11 it is noticed that the values of |1198853|

for normal force stiffness transverse force stiffness andthermal contact conductance for G-L theory are greaterthan the corresponding values for L-S theory It is seen that

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

7|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 8 Variation of |1198857| with angle of incidence (P-wave)

the values for ST1 are greater than the values for ST2 in therange 0∘ lt 120579

0lt 38∘ and in the further range the values for

ST2 are moreFigure 12 depicts that the values of amplitude ratios |119885

4|

for transverse force stiffness and thermal contact conduc-tance oscillate up to intermediate range and then decreasein the further range to attain minimum value at grazingincidence The values for ST1 ST2 NS1 and NS2 decreasein the whole range The values of |119885

4| for ST1 and ST2 are

magnified bymultiplying by a factor of 102 andNS1 NS2 TS1TS2 TCS1 and TCS2 are magnified by a factor of 10

Figure 13 shows that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is different The values of |1198855| for ST1

and ST2 are magnified by multiplying by a factor of 102 andNS1 NS2 TS1 TS2 TCS1 and TCS2 aremagnified by a factorof 10

From Figure 14 it is evident that the values of |1198856| for

all the boundary stiffnesses increase to attain maximumvalue and then decrease up to grazing incidence The valuesfor NS1 and TS1 are greater than the values for NS2 andTS2 respectively that reveals the thermal relaxation timeeffect The values of |119885

5| for ST1 and ST2 are magnified by

multiplying by a factor of 103 and NS1 NS2 TS1 TS2 TCS1and TCS2 are magnified by a factor of 102

Figure 15 depicts that the values of |1198857| for ST1 and ST2

increase in the interval 0∘ lt 1205790lt 39∘ and then decrease in

the further range It is noticed that there is slight differencein the values of amplitude ratio for L-S and G-L theory

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 9

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

0

02

04

Am

plitu

de ra

tio|Z

6|

08

06

1

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 7 Variation of |1198856| with angle of incidence (P-wave)

multiplying by 103 while the values for TCS1 and TCS2 aremagnified by multiplying by 102

Figure 8 depicts that the values of |1198857| for ST1 ST2 TS1

and TS2 increase in the range 0∘ lt 1205790lt 56∘ and decrease in

the remaining range The values for ST2 and TS2 are greaterthan the values for ST1 and TS2 respectively in the wholerange The values for normal force stiffness for G-L theoryremain more than the values for L-S theory The values of|1198857| for ST1 ST2 are magnified by multiplying by 108 the

values for TCS1 TCS2 are magnified by multiplying by 106and the values for NS1 NS2 TS1 and TS2 are magnified bymultiplying by 107

82 T-Wave Incident The values of amplitude ratio |1198851| for

transverse force stiffness and thermal contact conductancedecrease in the whole range with slight increase in the initialrange The values for ST1 and ST2 increase in the range 0∘ lt1205790lt 48∘ and then decreaseThese variations have been shown

in Figure 9 The values for TS1 TS2 TCS1 and TCS2 arereduced by dividing by 10

The values of amplitude ratio |1198852| for NS1 and NS2

increase in the whole range while the values for TS1 TS2ST1 ST2 TCS1 TCS2first decrease and then increase to attainmaximum value at grazing incidence These variations areshown in Figure 10

From Figure 11 it is noticed that the values of |1198853|

for normal force stiffness transverse force stiffness andthermal contact conductance for G-L theory are greaterthan the corresponding values for L-S theory It is seen that

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

7|

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 8 Variation of |1198857| with angle of incidence (P-wave)

the values for ST1 are greater than the values for ST2 in therange 0∘ lt 120579

0lt 38∘ and in the further range the values for

ST2 are moreFigure 12 depicts that the values of amplitude ratios |119885

4|

for transverse force stiffness and thermal contact conduc-tance oscillate up to intermediate range and then decreasein the further range to attain minimum value at grazingincidence The values for ST1 ST2 NS1 and NS2 decreasein the whole range The values of |119885

4| for ST1 and ST2 are

magnified bymultiplying by a factor of 102 andNS1 NS2 TS1TS2 TCS1 and TCS2 are magnified by a factor of 10

Figure 13 shows that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is different The values of |1198855| for ST1

and ST2 are magnified by multiplying by a factor of 102 andNS1 NS2 TS1 TS2 TCS1 and TCS2 aremagnified by a factorof 10

From Figure 14 it is evident that the values of |1198856| for

all the boundary stiffnesses increase to attain maximumvalue and then decrease up to grazing incidence The valuesfor NS1 and TS1 are greater than the values for NS2 andTS2 respectively that reveals the thermal relaxation timeeffect The values of |119885

5| for ST1 and ST2 are magnified by

multiplying by a factor of 103 and NS1 NS2 TS1 TS2 TCS1and TCS2 are magnified by a factor of 102

Figure 15 depicts that the values of |1198857| for ST1 and ST2

increase in the interval 0∘ lt 1205790lt 39∘ and then decrease in

the further range It is noticed that there is slight differencein the values of amplitude ratio for L-S and G-L theory

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Scholarly Research Notices

2

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 9 Variation of |1198851| with angle of incidence (P-wave)

1

08

06

04

0 10 20 30 40 50 60 70 80 90

ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

2|

Angle of incidence 1205790

Figure 10 Variation of |1198852| with angle of incidence (P-wave)

ST1NS1TS1TCS1ST2

NS2TS2TCS2

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790

Am

plitu

de ra

tio|Z

3|

Figure 11 Variation of |1198853| with angle of incidence (T-wave)

0

02

04

Am

plitu

de ra

tio|Z

4|

08

06

1

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 12 Variation of |1198854| with angle of incidence (T-wave)

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 11

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

0

02

04

Am

plitu

de ra

tio|Z

5|

08

06

1

Figure 13 Variation of |1198855| with angle of incidence (T-wave)

The values of |1198857| for ST1 and ST2 are magnified by multi-

plying by a factor of 108 and NS1 NS2 TS1 TS2 TCS1 andTCS2 are magnified by a factor of 106

83 SVWave Incident Variations of amplitude ratios |119885119894| 1 le

119894 le 7 with the angle of incidence 1205790 for incident SV-wave are

shown in Figures 16ndash22Figure 16 depicts that the values of |119885

1| for transverse

force stiffness and thermal contact conductance increase toattain peak values in the range 15∘ lt 120579

0lt 25∘ and then

decrease in the further range The values for ST1 and ST2increase in the range 0∘ lt 120579

0lt 35∘ and 45∘ lt 120579

0lt 66∘

respectively and decrease in the remaining range The valuesfor normal force stiffness increase from normal incidence toattain maximum value in the range 45∘ lt 120579

0lt 55∘ and then

decrease up to grazing incidence The values for TS1 TS2TCS1 and TCS2 are reduced by dividing by 10

Figure 17 shows that values of amplitude ratio |1198852| for

all the boundary stiffnesses follow oscillatory pattern in thewhole range The maximum value is attained by TCS1 in therange 15∘ lt 120579

0lt 25∘ It is seen that the values for L-S theory

are greater than the values for G-L theory in the whole rangeFigure 18 shows that the values of |119885

3| for transverse force

stiffness increase fromnormal incidence to grazing incidenceand attain peak value in the interval 15∘ lt 120579

0lt 25∘ The

values for normal stiffness and thermal contact conductancedecrease in the intervals 0∘ lt 120579

0lt 56∘ and 0∘ lt 120579

0lt 17∘

respectively and then increase in the further range

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

6|

06

04

02

0

Figure 14 Variation of |1198856| with angle of incidence (T-wave)

Figure 19 shows that the values of |1198854| for all the boundary

stiffnesses oscillate in the whole range The values for ST1 aregreater than the values for ST2 in the whole range exceptsome intermediate range The values of |119885

4| for ST1 and ST2

are magnified by multiplying by a factor of 103 and NS1NS2 TS1 and TS2 by a factor of 102 and TCS1 and TCS2 aremagnified by a factor of 10

Figure 20 depicts that the behavior of variation of |1198855| for

all the boundary stiffnesses is similar to that of |1198854| but the

magnitude of variation is differentThe values of |1198855| for ST1

ST2 are magnified by multiplying by a factor of 103 and NS1NS2 TCS1 TCS2 TS1 and TS2 are magnified by a factor of102

From Figure 21 it is noted that the values of |1198856| for

ST1 and ST2 decrease in the range 0∘ lt 1205790lt 45∘and

then increase with increase in angle of incidence The valuesof amplitude ratio for transverse force stiffness and thermalcontact conductance decrease in the whole range except theranges 16∘ lt 120579

0lt 24∘ and 17∘ lt 120579

0lt 23∘ respectively The

amplitude ratio for normal force stiffness attains maximumvalue at normal incidence The values of |119885

6| for ST1 ST2

are magnified by multiplying by a factor of 103 and NS1 NS2TCS1 TCS2 TS1 and TS2 are magnified by a factor of 102

Figure 22 depicts that the values of |1198857| for all the bound-

ary stiffnesses attain maximum value at the normal incidenceand then decrease with oscillation to attain minimum valueat the grazing incidence The values of |119885

7| for ST1 and ST2

are magnified by multiplying by a factor of 108 and NS1 NS2TS1 TS2 TCS1 and TCS2 are magnified by a factor of 106

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 International Scholarly Research Notices

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

16

12

08

04

0

Am

plitu

de ra

tio|Z

7|

Figure 15 Variation of |1198857| with angle of incidence for T-wave

16

12

08

04

0

Am

plitu

de ra

tio|Z

1|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 16 Variation of |1198851| with angle of incidence (SV-wave)

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

04

03

02

01

0

Am

plitu

de ra

tio|Z

2|

Figure 17 Variation of |1198852| with angle of incidence (SV-wave)

1

08

06

04

02

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

3|

Figure 18 Variation of |1198853| with angle of incidence (SV-wave)

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 13

16

12

08

04

0

Am

plitu

de ra

tio|Z

4|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 19 Variation of |1198854| with angle of incidence (SV-wave)

12

08

04

0

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Am

plitu

de ra

tio|Z

5|

Figure 20 Variation of |1198855| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

6|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 21 Variation of |1198856| with angle of incidence (SV-wave)

1

08

06

04

02

0

Am

plitu

de ra

tio|Z

7|

0 10 20 30 40 50 60 70 80 90

Angle of incidence 1205790ST1NS1TS1TCS1ST2

NS2TS2TCS2

Figure 22 Variation of |1198857| with angle of incidence (SV-wave)

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 International Scholarly Research Notices

9 Conclusion

Reflection and transmission at an interface between heatconducting elastic solid and micropolar fluid media arediscussed in the present paper Effect of normal force stiffnesstransverse force stiffness thermal contact conductance andthermal relaxation times is observed on the amplitude ratiosfor incidence of various plane waves (P-wave T-wave andSV-wave) When P-wave is incident it is noticed that thevalues of amplitude ratio for transverse force stiffness fortransmitted T-wave are greater than all the other boundarystiffnessesWhen plane wave (SV-wave) is incident the trendof variation of amplitude ratio for transmitted transversewave coupled with transverse microrotational wave that isC-I and C-II waves is similar but magnitude of oscillationis different The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value forG-L theory (when T-wave is incident)Themodel consideredis one of the more realistic forms of earth models and it maybe of interest for experimental seismologists in exploration ofvaluable materials such as minerals and crystal metals

Appendix

Consider the following

1198861119894= minus1198881119870119899

1205962

119881119894

radic1 minus1198812119894

11988120

sin21205790 119886

13= minus1198881119870119899

1205962

1198810

sin 1205790

11988614= minus

[1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790)]

1205962

11988121

+ 11988911989111989111198911+ 11989211198891198911

+ 1205801198881119870119899

120596

1198811

(radic1 minus11988121

11988120

sin21205790)

11988615= minus

[1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790)]

1205962

11988122

+ 11988911989111989111198911+ 11989221198891198911

+ 1205801198881119870119899

120596

1198812

(radic1 minus11988122

11988120

sin21205790)

11988616= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

11988617= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790+ 1205801198881119870119899

120596

1198810

sin 1205790

1198862119894= minus1198882119870119905

1205962

1198810

sin 1205790 119886

23= 1198882119870119905

1205962

1198813

radic1 minus11988123

11988120

sin21205790

11988624= (2119889119891

4+1198891198915)1205962

11988111198810

sin 1205790radic1minus

11988121

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988625= (2119889119891

4+1198891198915)1205962

11988121198810

sin 1205790radic1minus

11988122

11988120

sin21205790+1205801198882119870119905

120596

1198810

sin 1205790

11988626= 1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)+1198891198915

[[

[

radic1minus11988123

11988120

sin21205790minus1198913

]]

]

+ 1205801198882119870119905

120596

1198813

radic1 minus11988123

11988120

sin21205790

11988627= 1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)+ 1198891198915

[[

[

radic1minus11988124

11988120

sin21205790minus 1198914

]]

]

+ 1205801198882119870119905

120596

1198814

radic1 minus11988124

11988120

sin21205790

1198863119894= 1198884119870120579119891119894 119886

33= 0

11988634= [[

[

120580120596

1198811

radic1 minus11988121

11988120

sin21205790minus 1198884119870120579

]]

]

1198911

11988635= [[

[

120580120596

1198812

radic1 minus11988122

11988120

sin21205790minus 1198884119870120579

]]

]

1198912 119886

36= 11988637= 0

1198864119894= (1198891+ 1198892(1 minus

1198812119894

11988120

sin21205790))

1205962

1198812119894

+ (1 minus 1205911120580120596) 119891119894

11988643= 1198892

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

11988644

= minus[(1198891198912+ 1198891198913(1 minus

11988121

11988120

sin21205790))

1205962

11988121

+ (11988911989111989111198911+11988911989111198921)]

11988645

= minus[(1198891198912+ 1198891198913(1 minus

11988122

11988120

sin21205790))

1205962

11988122

+ (11988911989111989111198911+11988911989111198921)]

11988646= 1198891198913

1205962

11988131198810

sin 1205790radic1 minus

11988123

11988120

sin21205790

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 15

11988647= 1198891198913

1205962

11988141198810

sin 1205790radic1 minus

11988124

11988120

sin21205790

11988651= minus (2119889

4)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988652= minus (2119889

4)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988653= 1198894

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

11988654= minus (2119889119891

4+ 1198891198915)1205962

11988111198810

sin 1205790radic1 minus

11988121

11988120

sin21205790

11988655= minus (2119889119891

4+ 1198891198915)1205962

11988121198810

sin 1205790radic1 minus

11988122

11988120

sin21205790

11988656= minus[[

[

1198891198914

1205962

11988123

(1 minus 211988123

11988120

sin21205790)

+ 1198891198915(1205962

11988123

(radic1 minus11988123

11988120

sin21205790)minus 119891

3)]]

]

11988657= minus[[

[

1198891198914

1205962

11988124

(1 minus 211988124

11988120

sin21205790)

+ 1198891198915(1205962

11988124

(radic1 minus11988124

11988120

sin21205790)minus 119891

4)]]

]

1198866119894= 11988663= 11988664= 11988665= 0

11988666= 120580120596

1198813

1198913radic1 minus

11988123

11988120

sin21205790

11988667= 120580120596

1198814

1198914radic1 minus

11988124

11988120

sin21205790

1198867119894= 120580120596

119881119894

radic1 minus1198812119894

11988120

sin21205790119891119894 119886

73= 0

11988674= 1205801199012

120596

1198814

radic1 minus11988121

11988120

sin212057901198911

11988675= 1205801199012

120596

1198815

radic1 minus11988122

11988120

sin212057901198912

11988676= 11988677= 0 (119894 = 1 2)

1198891=120582

12058811988821

1198892=2120583

12058811988821

1198893=1198892

2

1198891198911198911=119887

] 119889119891

1=

1198880

120588]1198790

1198891198912=120582119891120596lowast

12058811988821

1198891198913=(2120583119891 + 119870119891) 120596lowast

12058811988821

1198891198914=120583119891120596lowast

12058811988821

1198891198915=119870119891120596lowast

12058811988821

1199012=119870lowast1

119870lowast

1198881= 1198882=]1198790

12058811988821

1198884=

]11988821

120596lowast119870lowast1

(A1)

Symbols

120582 120583 Lamersquos constants119905119894119895 Components of the stress tensor

Displacement vector120588 Density119870lowast Thermal conductivity119888lowast Specific heat at constant strain1198790 Uniform temperature

119879 Temperature change120572119879 Coefficient of linear thermal expansion

120575119894119895 Kronecker delta

120576119894119895119903 Alternating symbol

120582119891 120583119891 119870119891120572119891 120573119891 120574119891 119888

0 Material constants of the fluid

120590119891119894119895 Components of stress tensor in the fluid

119898119891119894119895 Components of couple stress tensor in the

fluidV Velocity vectorΨ Microrotation velocity vector120588119891 Density119868 Scalar constant with the dimension of

moment of inertia of unit mass119901 Pressure119870lowast1 Thermal conductivity

1198861198791198910 Specific heat at constant strain

1198791198910 Absolute temperature

119879119891 Temperature change120601lowast119891 Variation in specific volume120572119879119891 Coefficient of linear thermal expansion

Conflict of Interests

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

Acknowledgment

Thanks are due to Punjab Technical University Kapurthalafor providing research facilities and enrolling one of the

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 International Scholarly Research Notices

authors (Mandeep Kaur) as a research scholar to PhDProgramme

References

[1] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 pp 205ndash217 1964

[2] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of AppliedMathematics and Mechanics vol 16 pp 1ndash18 1966

[3] T Ariman M A Turk and N D Sylvester ldquoMicrocontinuumfluid mechanics-A reviewrdquo International Journal of EngineeringScience vol 11 no 8 pp 905ndash930 1973

[4] T Ariman N D Sylvester and M A Turk ldquoApplicationsof microcontinuum fluid mechanicsrdquo International Journal ofEngineering Science vol 12 no 4 pp 273ndash293 1974

[5] P Rıha ldquoOn the theory of heat-conducting micropolar fluidswith microtemperaturesrdquo Acta Mechanica vol 23 no 1-2 pp1ndash8 1975

[6] A C Eringen and C B Kafadar ldquoPolar field theoriesrdquo inContinuum Physics A C Eringen Ed vol 4 Academic PressNew York NY USA 1976

[7] O Brulin Linear micropolar media in Mechanics of MicropolarMedia edited by O Brulin and R K T Hsieh World ScientificSingapore 1982

[8] R S R Gorla ldquoCombined forced and free convection in theboundary layer flow of a micropolar fluid on a continuousmoving vertical cylinderrdquo International Journal of EngineeringScience vol 27 no 1 pp 77ndash86 1989

[9] A C Eringen ldquoTheory of thermo-microstretch fluids andbubbly liquidsrdquo International Journal of Engineering Science vol28 no 2 pp 133ndash143 1990

[10] N U Aydemir and J E S Venart ldquoFlow of a thermomicropolarfluid with stretchrdquo International Journal of Engineering Sciencevol 28 no 12 pp 1211ndash1222 1990

[11] M Ciarletta ldquoSpatial decay estimates for heat-conductingmicropolar fluidsrdquo International Journal of Engineering Sciencevol 39 no 6 pp 655ndash668 2001

[12] S Hsia and J Cheng ldquoLongitudinal plane wave propagation inelastic-micropolar porous mediardquo Japanese Journal of AppliedPhysics vol 45 no 3 pp 1743ndash1748 2006

[13] S Y Hsia S M Chiu C C Su and T H Chen ldquoPropagationof transverse waves in elastic-micropolar porous semispacesrdquoJapanese Journal of Applied Physics vol 46 no 11 pp 7399ndash7405 2007

[14] S K Tomar and M L Gogna ldquoReflection and refraction ofa longitudinal microrotational wave at an interface betweentwo micropolar elastic solids in welded contactrdquo InternationalJournal of Engineering Science vol 30 no 11 pp 1637ndash16461992

[15] S K Tomar and M L Gogna ldquoReflection and refractionof longitudinal wave at an interface between two micropolarelastic solids in welded contactrdquo Journal of the Acoustical Societyof America vol 97 no 2 pp 822ndash830 1995

[16] S K Tomar and M L Gogna ldquoReflection and refraction ofcoupled transverse and micro-rotational waves at an interfacebetween two different micropolar elastic media in weldedcontactrdquo International Journal of Engineering Science vol 33 no4 pp 485ndash496 1995

[17] R Kumar N Sharma and P Ram ldquoReflection and transmissionof micropolar elastic waves at an imperfect boundaryrdquo Multi-discipline Modeling in Materials and Structures vol 4 no 1 pp15ndash36 2008

[18] R Kumar N Sharma and P Ram ldquoInterfacial imperfectionon reflection and transmission of plane waves in anisotropicmicropolar mediardquoTheoretical and Applied Fracture Mechanicsvol 49 no 3 pp 305ndash312 2008

[19] D Singh and S K Tomar ldquoLongitudinal waves at a micropolarfluidsolid interfacerdquo International Journal of Solids and Struc-tures vol 45 no 1 pp 225ndash244 2008

[20] Q Sun H Xu and B Liang ldquoPropagation characteristicsof longitudinal displacement wave in micropolar fluid withmicropolar elastic platerdquoMaterials Science Forum vol 694 pp923ndash927 2011

[21] H Y Xu S Y Dang Q Y Sun and B Liang ldquoReflection andtransmission characteristics of coupled wave through microp-olar elastic solid interlayer in micropolar fluidrdquo InternationalJournal of Automation and Power Engineering vol 2 no 6 pp347ndash354 2013

[22] S Y Dang H Y Xu Q Y Sun and B Liang ldquoPropagation char-acteristics of coupled wave through micropolar fluid interlayerin micropolar elastic solidrdquo Advanced Materials Research vol721 pp 729ndash732 2013

[23] C Fu and P J Wei ldquoThe wave propagation through theimperfect interface between two micropolar solidsrdquo AdvancedMaterials Research vol 803 pp 419ndash422 2013

[24] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Stiintifice ale Univer-sitatii Ovidius Constanta vol 22 pp 151ndash175 2014

[25] S K Vishwakarma S Gupta and A K Verma ldquoTorsionalwave propagation in Earthrsquos crustal layer under the influence ofimperfect interfacerdquo Journal of Vibration and Control vol 20no 3 pp 355ndash369 2014

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

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

[28] A N Sinha and S B Sinha ldquoReflection of thermoelastic wavesat a solid half-space with thermal relaxationrdquo Journal of Physicsof the Earth vol 22 no 2 pp 237ndash244 1974

[29] S B Sinha and K A Elsibai ldquoReflection of thermoelastic wavesat a solid half-space with two relaxation timesrdquo Journal ofThermal Stresses vol 19 no 8 pp 749ndash762 1996

[30] H Deresiewicz ldquoEffect of boundaries on waves in a thermoe-lastic solid reflexion of plane waves from a plane boundaryrdquoJournal of the Mechanics and Physics of Solids vol 8 pp 164ndash172 1960

[31] R S Dhaliwal andA SinghDynamic CoupledThermoelasticityHindustan Publication Corporation New Delhi India 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of