effect of loose bonding and corrugated boundary surface on ...t300/5208 graphite/ epoxy composite by...

LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e01

AbstractTheproblemsconcernstothepropagationofsurfacewavepropaga‐tionthroughvariousanisotropicmediumswithinitialstressandirreg‐ularboundariesareofgreatinteresttoseismologists,duetotheirap‐plicationstowardsthestabilityofthemedium.ThepresentpaperdealswiththepropagationofRayleigh‐typewaveinacorrugatedfibre‐rein‐forcedlayerlyingoveraninitiallystressedorthotropichalf‐spaceun‐dergravity.Theupperfreesurfaceisassumedtobecorrugated;whiletheinterfaceofthelayerandhalf‐spaceiscorrugatedaswellaslooselybonded.Thefrequencyequationisdeducedinclosedform.Numericalcomputationhasbeencarriedoutwhichaidstoplotthedimensionlessphasevelocityagainstdimensionlesswavenumberforsakeofgraph‐icaldemonstration.Numericalresultsanalyzetheinfluenceofcorruga‐tion,loosebonding,initialstressandgravityonthephasevelocityofRayleigh‐typewave.Moreover,thepresenceandabsenceofcorruga‐tion,loosebondingandinitialstressisalsodiscussedincomparativemanner.KeywordsRayleigh‐typewave,initialstress,gravity,looselybonded,corruga‐tion,fibre‐reinforced,orthotropic

EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

NOMENCLATURE

H Theaveragewidthofthelayer( )if x Periodicfunctionsandindependentof y . ,j jl lR I ThecosineandsineFouriercoefficientsrespectively.

b Thewavenumberassociatedwithcorrugatedboundarysurfaces.

1 2,a a Theamplitudesofcorrugation.

1 2 3, ,u u u Thedisplacementcomponentsoffibre‐reinforcedlayeralong , ,x y z directionrespectively.* * *1 2 3, ,u u u Thedisplacementcomponentsoforthotropichalf‐spacealong , ,x y z directionrespectively.

* * *1 2 3, ,a a a a

thepreferreddirectionofreinforcement.

ij Thecomponentsofstressoflayer.

ije Thecomponentsofinfinitesimalstrainoflayer.

ij Kroneckerdelta.

T Transverseshearmodulusoflayer.

L Longitudinalshearmodulusoflayer., Specificstresscomponentsforconcretepartofthecompositematerial.

Lame’sconstantofelasticity.P Initialcompressionin x ‐direction.

1 2, Mediumdensity,

ij Thestresscomponentsofhalf‐space.

AbhishekKumarSinghaKshitishCh.Mistria,*MukeshKumarPalaaDepartmentofAppliedMathematics,IIT ISM ,Dhanbad‐826004Email:abhi.5700@gmail.com,kchmistri@gmail.com,3007mukesh@gmail.com*Correspondingauthorhttp://dx.doi.org/10.1590/1679‐78253577Received02.12.2016Inrevisedform28.09.2017Accepted29.09.2017Availableonline30.09.2017

Original Article

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e01 2/15

ij Therotationalcomponentsofhalf‐space.

ijC Stiffnesstensorcomponentsincontractionnotation.

G Biot’sgravityparameter. Bondingparameter.1 INTRODUCTION

Theproblemsofelastodynamicsarenotlimitedtothemechanicsofthoseelasticmaterialswhicharesimplyisotropic,rathertheproblemstakeamoregeneralandrealisticformwhenthemediaconsideredareanisotropic.Thepresenceofsomeeffectivephysicalfactorsnamelyinitialstress,hydrostaticstress,cracks,fractures,etc.causesthemediumstobe‐haveanisotropicallytothepropagationofwavesthroughit.Theseinitialstresses tensile/compressive aretheresultsofoverburdenedlayer,atmosphericpressure,variationintemperature,slowprocessofcreepandgravitationalfield.Tensilestressissaidtoresponsibleformorerigidityandcompressivestressforlessrigidityofamedium.Ontheotherhand,presenceoffibre‐reinforcedmaterialsinearth’scrust,intheformofhardorsoftrocksmayalsoaffectthewavepropagation.Thesecompositematerialsadoptself‐reinforcedbehaviorundercertaintemperatureandpressure.Itfindsnumerousapplicationsinconstruction,civilengineering,geophysicsandgeomechanicsduetoitslowweightandhighstrength.Thereinforcementofsoil,bothnaturallyandsynthetically,enhancesthestrengthandloadbearingcapacityofit.Themechanicalbehaviorofcompositematerialscouldbewellunderstoodthroughthestudyofanisotropicelasticity.Carbon,nylonorconceivablemetalwhiskers,etc.aregoodmodelsoffibre‐reinforcedmaterials.PrikazchikovandRog‐erson 2003 studiedtheeffectofpre‐stressonthepropagationofsmallamplitudewavesinanincompressible,trans‐versely isotropic elastic solid. Prosser and Green 1990 calculated some of the nonlinear third order moduli ofT300/5208graphite/epoxycompositebymeasuringthenormalizedchangeinultrasonic"natural"velocityasafunctionofstressandtemperature.AlotofinformationaboutsuchreinforcedmaterialscanbegainedfromSpencer 1972 whoanalysesthemacroscopicpropertiesoffiber‐reinforcedmaterials.Intherecentpast,ChattopadhyayandSingh 2012 studiedthepropagationofhorizontallypolarisedshearwavesinaninternalirregular rectangularandparabolicirreg‐ularity magnetoelasticself‐reinforcedstratumsandwichedbetweentwosemi‐infinitemagnetoelasticself‐reinforcedmedia.SomemoreimportantworksincludeFanandHwu 1998 ,Grünewaldetal. 2012 ,ChattopadhyayandSingh2013 ,Chattopadhyayetal. 2010 ,SamalandChattaraj 2011 ,Sethietal. 2016 ,Abd‐Alla 1999 andGaurandRana2014 .

Anotherimportantclassofmaterialwhichmaybeconsideredinthestudyofelastodynamicproblemsistheor‐thotropicmaterial.Themechanicalpropertiesofsuchmaterialsareuniqueandindependentinthreemutuallyper‐pendiculardirections.Sometimessomefiber‐reinforcedcompositesimitateorthotropicmaterials.

ChaiandWu 1996 extendedtheBarnett‐Lothe'sintegralformalisminordertodeterminethevelocitiesofsur‐facewavespropagatinginapre‐stressedanisotropiccrystal.SinghandYadav 2013 dealtwiththereflectionofqPandqSVwavesatafreesurfaceofaperfectlyconductingtransverselyisotropicelasticsolidhalf‐spaceunderinitialstress.UsingRayleigh'smethodofapproximation,thereflectionandtransmissionofplaneqP‐waveatacorrugatedinterfacebetweentwodissimilarpre‐stressedelasticsolidhalf‐spaceswasdiscussedbySinghandTomar 2008 .AtremendousamountofknowledgecanbegatheredregardingtheeffectofgravityonthepropagationofwavesfromBiot 1965 ;andalsothroughsomepapersincludingDasetal. 1992 andChattopadhyayetal. 2009 .Moreover,KumarandKumar 2011 ,Destrade 2001 andChow 1971 havealsocontributedconsideringanorthotropicmate‐rialmediumintheirstudy.

Changingmediummayaffectthewavepropagation;intimatingthattheboundarysurfaces freeboundaryorin‐terface ofmediumsplayimportantroleinthestudyofwavepropagationthroughdifferentmediums.Itisnotneces‐sarythattheboundarysurfacealwaysacquirearegularplanarshape.Whiledealingwithdifferentelastodynamicalproblems,onemayencounterboundarysurfacesofdifferentshapes.Forexample,theboundariesmaypossessarec‐tangularorparabolicirregularity;oritmaybecorrugated.Corrugatedboundarysurfacemaybedefinedasaseriesofparallelridgesandfurrows.Theundulatoryfactorofsuchboundariesaffectsthepropagationofwavesandvibrations.Further,theinterfaceoftwomediumsmaynotbealwaysweldedratheritmaybelooselyboundedtoo.

Thestudyofcorrugatedboundarysurfacesandlooselybondedinterfacesofmaterialmediumsisalsoimportanttounderstandthebehaviorofwavepropagation.StartingwithTomarandKaur 2007 ,Singh 2011,2014 ,SinghandKumar 1998 ,KhuranaandVashisth 2001 ,continuedtoNandalandSaini 2013 andSinghetal. 2015 hadstudiedthepropagationofwavesthroughcorrugatedboundarysurfacesandlooselybondedinterfaces.

ThecurrentstudyinvestigatesthepropagationofRayleigh‐typewaveinafibre‐reinforcedlayeroverlyinganin‐itiallystressedorthotropichalf‐spaceundergravity.Theupperfreesurfaceisassumedtobecorrugated;whiletheinterfaceofthelayerandhalf‐spaceiscorrugatedaswellaslooselybonded.Theclosedformexpressionoffrequencyequationisderivedandnumericalcomputationforphasevelocityisperformedwhichisreflectedgraphically.Theeffectofcorrugation, loosebonding, initialstressandgravityonthephasevelocityofRayleigh‐typewaveishigh‐lightedinthestudy.

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

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2 FORMULATIONANDSOLUTIONOFTHEPROBLEM

ConsiderthepropagationofRayleigh‐typewaveinafibre‐reinforcedlayerlyingoveraninitiallystressedorthotropichalf‐spaceundergravitywheretheupperfreesurfaceiscorrugatedandtheinterfaceofthelayerandhalf‐spaceiscorrugatedaswellaslooselybonded.TheaveragewidthofthelayerisassumedtobeH .Cartesianco‐ordinatesystemischoseninsuchawaythat x ‐axisisthedirectionofwavepropagation, z ‐axisispositivelypointingdownwardsandtheoriginisattheinterfaceofthelayerandhalf‐space.ThesaidgeometryisillustratedinFig.1.

Lettheequationofuppermostcorrugatedboundarysurfacebe 2 ( )z f x H andtheequationofcorrugatedin‐terfacebetweenlayerandhalf‐spacebe 1 ( ),z f x where 1 ( )f x and 2 ( )f x areperiodicfunctionsandindependentofy .TakingasuitableoriginofcoordinateswecanrepresentTrigonometricFourierseriesof. 1 ( )f x ., 2 ( )f x asfollowsAsano, 1966 :

( ) ( )

1

, 1,2,j ilbx j ilbxj l l

l

f f e f e j

1

where ( )jlf and ( )j

lf areFourierexpansioncoefficientsand l isseriesexpansionorder.Letusintroducetheconstants

1 2, , ,j jl la a R I asfollows:

(1) (2) ( )1 21 1, , , 1,2, and 2,3,...

2 2 2

j jj l l

l

R iIa af f f j l

and

1 1(1) (1)1 1 2 2cos cos2 sin 2 ... cos sin ...,l lf a bx R bx I bx R lbx I lbx

2 2(2) (2)2 2 2 2cos cos2 sin 2 ... cos sin ...,l lf a bx R bx I bx R lbx I lbx

where ,j jl lR I arethecosineandsineFouriercoefficientsrespectively.Asfarthepresentproblemisconcerned,the

corrugatedupperboundarysurfaceandlowerboundarysurfacemaybeexpressedwiththeaidofcosinetermsi.e.

1 1 cosf a bx and 2 2 cosf a bx respectively,whereb isthewavenumberassociatedwithcorrugatedboundarysur‐faces, 1a and 2a aretheamplitudesofcorrugationandthewavelengthofthecorrugationis 2 / b .

Figure1:Typicalstructureofanalysis.

Letusassume 1 2 3, ,u u u and * * *1 2 3, ,u u u

asthedisplacementcomponentsofupperfibre‐reinforcedlayerand

lowerinitiallystressedorthotropichalf‐spaceundergravityrespectively.3 GOVERNINGEQUATIONSANDSOLUTIONOFTHEPROBLEM

ForthepropagationofRayleigh‐typewave,weconsider

1 1 2 3 3

* * * * *1 1 2 3 3

( , , ), 0, ( , , ),

( , , ), 0, ( , , ),

u u x z t u u u x z t

u u x z t u u u x z t

2

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

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andtheconditionforplainstraindeformationin xz ‐planeis 0.y

Herewedenotethepartialderivativewithrespect toavariable 1,2,3ix i by i .Thefirstandsecondtime

derivativearerepresentedas t and tt respectively.Moreover, xd and xxd standsford

dx and

2

2

d

dxrespectively.

3.1DynamicsofFibre‐ReinforcedMaterial

Theconstitutiveequation fora fibre‐reinforced linearlyelasticanisotropicmediumwithpreferreddirection a is

givenby Spencer 1972

* * * * * * * * * * * *2 2 ,

, , , 1,2,3.

ij kk ij T ij k m km ij i j kk L T i k kj j k ki k m km i je e a a e a a e a a e a a e a a e a a

i j k m

3

where ij arethecomponentsofstress, 1

2ij j i i je u u arethecomponentsofinfinitesimalstrain, ij isKronecker

delta, * * *1 2 3, ,a a a a

, which may be function of position, is the preferred direction of reinforcement such that

3 2*

1

1ii

a

.Indicestakethevalues1,2,3andsummationconventionisemployed. , and L T arereinforce‐

mentparameters. T and L canbeidentifiedasthetransverseshearandlongitudinalshearmodulusinthepre‐ferreddirectionsrespectively. ,

arespecificstresscomponentstotakeintoaccountdifferentlayersforconcrete

partofthecompositematerial, isLame’sconstantofelasticity.Now,theequationsofmotionwithoutbodyforceare

, 1ij j tt iu 4

where 1 standsformassdensity.UsingEquations 2 and 3 ,Equation 4 reducesto

1 1 1 3 1 1 1 1,xx xz zz ttP u Q u R u u 5

1 3 1 1 2 3 1 3 ,xx xz zz ttR u Q u P u u 6where

1 1 2 12 4 2 , , 2 , .L T L T LP Q P R

AssumethesolutionofEquations 5 and 6 as

( )1 1

kz ik x ctu A e , ( )3 3 ,kz ik x ctu A e 7

where 1 3( , 0, ) aretheunitdisplacementvectorcomponents.InviewofEquation 7 ,Equations 5 and 6 yields

2 21 1 1 1 3( ) 0,L c P i Q 8

2 2

1 1 2 1 1 3( ) 0.i Q P c R 9

Fornon‐trivialsolutionofEquations 8 and 9 ,itfollowsthat

4 22 0,LP m n 10

where

222

2

4,

2L

L

m m P n

P

2 2 22 1 1 1( ) ( ) ,L Lm c P c P Q

2 21 1 1( )( )Ln c c R ,

11

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

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sothat,Equations 7 and 8 gives

2 21 13

1 1

, 1,2j Lj

j

c Puj

u i Q

.

Therefore, thedisplacementcomponentsoftheupperFibre‐reinforcedlayerforpropagationofRayleigh‐typewavearefoundas

1 2 1 2 ( )1 1 2 3 4( ) ,k z k z k z k z ik x ctu Ae A e A e A e e 12

1 1 2 2 ( )3 1 1 3 2 2 4( ) ( ) .k z k z k z k z ik x ctu Ae A e A e A e e 13

3.2DynamicsoftheLowerInitiallyStressedOrthotropicHalf‐SpaceunderGravity

Thedynamicalequationsofmotionforanelasticmediumundergravityandinitialcompressionstress P in x ‐direc‐tionare

11 12 13 12 13 2 3 2 1 ,x y z y z x ttP g u u 14

12 22 23 12 2 2 ,x y z x ttP u 15

13 23 33 12 2 1 2 3 ,x y z x x ttP g u u 16

where 2 isthemediumdensity, ij arethestresscomponentsand 1

2 jij jiiu u aretherotationalcomponents.

ForthepropagationofRayleigh‐typewave,Equations 14 , 15 and 16 withtheaidofEquation 2 ,leadsto

11 13 13 2 3 2 1 ,x z z x ttP g u u 17

13 33 13 2 1 2 3 .x z x x ttP g u u 18

Thestresscomponents ij inthiscasecanbewrittenas:

11 11 1 13 3 ,x zC P u C P u 19

22 13 1 33 3 ,x zC u C u 20

13 11 13 1 3

1,

2 z xC C u u 21

where ijC arestiffnesstensorcomponentsincontractionnotation.Since,theproblemisconfinedto xz ‐planeonly,it

isfoundthat

12 22C C 23 0C .

SubstitutingEquations 19 , 20 , 21 andtakingintoconsiderationtheaboveassumptions,Equations 17 and18 canberewrittenintermsofthedisplacementcomponents 1 3,0,u u as

11 1 1 3 13 3 1 2 3 2 12 2 2 ,xx zz xz xz zz x ttC P u u u C u u g u u 22

11 1 3 13 1 3 33 3 2 1 2 32 2 2 .xz xx xz xx zz x ttC u u C P u u C u g u u 23

Thedisplacements 1u and 3u canbederivedfromthedisplacementpotentials ( , , ) andx y t ( , , )x y t usingtherelations

1 3, .x z z xu u 24

Equations 22 and 23 whensubstituteduponbyEquation 24 ,respectivelygive

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e01 6/15

211 2 2 ,x ttC P g 25

211 13 2 22 x ttC C P g 26

and

11 33 2 2 ,xx zz x ttC C g 27

211 13 33 2 22 2 2 ,xx zz zz x ttC C P C g 28

where

2

.xx zz

Itmaybenotedthat,asthedirectionofinitialcompressivewaveistakenalong x ‐axis,thebodywavevelocitiesmustbedifferentin x and z directions.Thus,onlyEquations 25 and 28 istobeconsideredwithaviewthatthewaveispropagatinginthedirectionof x only.Equations 25 and 28 correspondtocompressiveandshearwaverespectively,alongthe x directiononly;whileEquations 26 and 27 correspondtocompressiveandshearwaverespectivelyalongthe z directiononly.

InordertosolvetheEquations 25 and 28 ,itisassumedthat

,ik x ctz e 29

.

ik x ct

z e

30

UsingEquations 29 and 30 inEquations 25 and 28 itisobtainedthat

2 2 221

0 , zz

k gd k M i

31

2 22

1

0 , zz

k gd k N i

32

where

222 2 2 11 132

2 21 1

21 , ,

c C C PcM N

2 21 11 1 33 11 13, 2 .C P C C C P

33

Equation 31 and 32 suggeststhat

2 21 2 , 0,zz zzd k s d k s 34

where

2

2 2 2 2 2 2 2 2 441 2 1 2

1 1

, .GC

s s M N s s M N

35

and 2

44

gG

C k

isBiot’sgravityparameter.

Now,weassumethesolutionofEquation 34 oftheform

1 2 1 2* * * *5 6 7 8 ,iks z iks z iks z iks zz A e A e A e A e 36

where * * * *5 6 7 8, , ,A A A A arearbitraryconstants.

Keepinginviewthatthedisplacementvanishesas z ,theappropriatesolutionofEquations 25 and 28 maybewrittenas

1 25 6 ,ik x ctiks z iks zA e A e e 37

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

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1 25 6 ,ik x ctiks z iks zA e A e e 38

wherethearbitraryconstants 5 6,A A arerelatedto 5 6,A A respectivelybymeansofEquations 31 or 32 .

Thecoefficientof 1iks ze and 2iks ze whenequatedtozero,Equation 31 gives

5 1 5 6 2 6, ,A ikk A A ikk A

where

2 2 21 44

, 1, 2.j

j

s M kCk j

G

39

Therefore,theexpressionsforthedisplacementpotentialsare

1 25 6 ,ik x ctiks z iks zA e A e e 40

1 25 1 6 2 .ik x ctiks z iks zik A k e A k e e 41

Sothatthedisplacementcomponentsmaybewrittenas

1 22 21 5 1 1 6 2 2 .ik x ctiks z iks zu A ik k s k e A ik k s k e e 42

1 3 2 32 23 5 1 1 6 2 2 .ik x ctiks x iks xu A iks k k e A iks k k e e 43

4 BOUNDARYCONDITIONSANDSOLUTIONOFTHEPROBLEM

Followingaretheboundaryconditionsattheuppermostcorrugatedsurface,andatthecommoncorrugatedaswellaslooselybondedinterfaceoflayerandhalf‐space:

i Tractionfreeconditionattheuppersurface:

33 2 31 13 2 11 20, 0, atf x f x z f x H 44

ii Conditionforcontinuityofstressesatthecommoninterface

33 1 31 33 1 31 13 1 11 13 1 11 1, , atf x f x f x f x z f x 45

iii Conditionforcontinuityofnormaldisplacementsatthecommoninterface

3 3 1, atu u z f x 46

iv Conditionfortheproportionalityofshearstresstotheslipatthecommoninterface Vashisthetal. 1991

113 1 11 44 1 1 1, at

1 T

f ikC c u u z f x

47

where 0 1 isthebondingparameter.Thecommoninterfaceissaidtobeperfectlybondedforthecasewhen

1 ;andideallysmoothwhen 0 .Substitutingtheexpressionsoftheobtaineddisplacementcomponents,asgiveninEquations 12 , 13 , 42 and

43 ,intheaboveboundaryconditions,sixhomogeneousequationsin jA 1,2...,6j areobtainedwhosenon‐trivial

solutionrequires

0ijt , 48

whereentries ijt areasprovidedintheAppendix.Equation 48 isthedispersionrelationforthepropagationofRay‐

leigh‐typewaveinacorrugatedfibre‐reinforcedlayerlyingoveraninitiallystressedorthotropichalf‐spaceundergravity.

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5 NUMERICALRESULTSANDDISCUSSION

ThenumericalvalueswhichhavebeentakenintoconsiderationwithaviewtoperformnumericalcomputationofphasevelocityofRayleigh‐typewavepropagatinginacorrugatedfibre‐reinforcedlayerlyingoveraninitiallystressedorthotropichalf‐spaceundergravity,withlooselybondedcommoninterface,areasfollows:

Forfibre‐reinforcedlayer Markham, 1970

9 2 9 2 9 2 9 27.07 10 N/ m , 3.5 10 N/ m , 1.28 10 N/ m , 220.90 10 N/ m ,L T 9 2 3

15.66 10 N/ m , 1600kg/m .

Forinitiallystressedorthotropichalf‐spaceundergravity ProsserandGreen 190

9 2 9 2 9 2 9 211 13 33 4414.295 10 N/ m , 9 10 N/ m , 108.4 10 N/ m , 5.28 10 N/ m ,C C C C

32 1442kg/m .

Figs.1to6irradiatetheeffectsofundulation,corrugation,bondingofthelayerandhalf‐space,initialstressand

gravityonthephasevelocityofRayleigh‐typewavepropagatinginafibre‐reinforcedlayerlyingoveranorthotropic

half‐space.Inallthefigures,dimensionlessphasevelocity 1 Tc hasbeenplottedagainstdimensionlesswave

number kH fordifferentvaluesoftheaffectingparameters.ThesefiguressuggestthatthephasevelocityofRay‐

leigh‐typewavedecreaseswithincreaseinwavenumber.1.EffectofCorrugatedBoundarySurfacesonthePhaseVelocityofRayleigh‐TypeWave

Figs.2,3and4showtheeffectofcorrugationandundulationonthephasevelocityofRayleigh‐typewave.Inparticu‐lar,Fig.2showstheeffectofcorrugationparameterassociatedwiththeupperboundarysurface 1a b ;Fig.3inter‐

pretstheeffectofcorrugationparameterassociatedwiththecommoninterfaceoflayerandhalf‐space 2a b ;andFig.

4showstheeffectofundulationparameter bH andpositionparameter x H ,onthephasevelocityofRayleigh‐

typewave.Curve1inFigs.2and3correspondtothecaseswhenthereisnocorrugationintheupperboundarysurfaceandthecommoninterfaceoflayerandhalf‐spacerespectively.Figs.2and3elucidatethatthephasevelocityofRay‐leigh‐typewaveincreaseswithincreaseinthemagnitudeofcorrugationparameterassociatedwiththeupperbound‐arysurfacewhereasitdecreaseswithincreaseinthemagnitudeofcorrugationparameterassociatedwiththecom‐moninterfaceoflayerandhalf‐space.ComparativestudyofFigs.2and3suggestthattheabsenceofcorrugationinupperboundarysurfacedisfavorsthephasevelocity;buttheabsenceofcorrugationattheinterfaceofthelayerandhalf‐spacegreatlysupportsthephasevelocity.ThesimilarantagonisticbehaviorofcorrugationonthephasevelocityofRayleigh‐typewaveatupperboundarysurfaceandcommoninterfaceoflayerandhalf‐spaceismarkedbySinghetal 2016,2017 .Fig.4manifeststhatundulationparameteralongwithpositionparameteralsohasgreatimpactonthephasevelocityofRayleigh‐typewaveasasmallincreaseintheirmagnitudesignificantlyincreasesthephaseve‐locity Singhetal.,2017 .

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

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Figure2:Variationofphasevelocity 1 Tc withwavenumber kH fordifferent

valuesofcorrugationparameterassociatedwiththeupperboundarysurface 1a b .

Figure3:Variationofphasevelocity 1 Tc withwavenumber kH fordifferent

valuesofcorrugationparameterassociatedwiththeinterface 2a b .

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e01 10/15

Figure4:Variationofphasevelocity 1 Tc withwavenumber kH fordifferentvalues

ofundulationparameter bH andpositionparameter x H .

2.EffectofInitialStressonthePhaseVelocityofRayleigh‐TypeWave

Theeffectofinitialstress 442P C onthephasevelocityofRayleigh‐typewaveisdemonstratedinFig.5.Inthisfigure,

curves1and2correspondtothecasewhenthehalf‐spaceisunderhorizontaltensileinitialstress 442 0P C ;curve

3representthecasewhenhalf‐spaceisundernoinitialstress 442 0P C ;andcurves4and5correspondtothecase

whenhalf‐spaceisunderhorizontalcompressiveinitialstress 442 0P C .Itcanbeobservedfromthefigurethat

thephasevelocityofRayleigh‐typewaveincreaseswithincreaseininitialstress.MeticulousexaminationofthefigureconcludesthatthephasevelocityofRayleigh‐typewavedecreaseswithincreaseinthemagnitudeofhorizontaltensileinitialstress;whereasitincreaseswithincreaseinthemagnitudeofhorizontalcompressiveinitialstress.Moreover,theinfluenceofinitialstressinfoundsignificantatlowfrequencyregionascomparetothehighfrequencyregion.Similarresultmaybeobservedwhenthecaseofgravitytendingtozeroi.e. 0G istakeninthestudybyAbd‐Alla1999 .3.EffectofLooseBondingonthePhaseVelocityofRayleigh‐TypeWave

Theinfluenceoflooselybondedinterfaceofthelayerandhalf‐spaceonthephasevelocityofRayleigh‐typewaveismarkedbyplottingthedispersioncurvefordifferentvaluesofbondingparameter, whichhasbeendemonstratedinFig.6.Curve1inthefigureinterpretsthecasewhentheinterfaceoflayerandhalf‐spaceareneartoasmoothcontact 0.01 ;curves2,3and4representthattheyarelooselybonded 0 1 ;andcurve5correspondstothecase

whentheinterfaceisperfectlybonded inweldedcontact 1 .Thefiguremanifeststhatthephasevelocityde‐

creaseswithincreaseinthemagnitudeofbondingparameter.Minuteobservationofthefiguresetforththefactthat,thevariationofbondingparameterfromloosebondingtowardssmoothcontactmildlyaffectsthephasevelocityofRayleigh‐typewaveincomparisontoitsvariationfromloosebondingtowardsweldedcontact.

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Figure5:Variationofphasevelocity 1 Tc withwavenumber kH

fordifferentvaluesofinitialstressparameter 442P C .

Figure6:Variationofphasevelocity 1 Tc withwavenumber kH

fordifferentvaluesofbondingparameter .

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Figure7:Variationofphasevelocity 1 Tc withwavenumber kH

fordifferentvaluesofBiot’sgravityparameter G .

4.EffectofGravityonthePhaseVelocityofRayleigh‐TypeWave

TheeffectofgravityonthephasevelocityofRayleigh‐typewaveisshowninFig.7.Curve1inthisfigurerepresentsthecasewheneffectofgravityisneglected 0G whereascurve2,3,4,5correspondstothecasewheneffectof

gravityinincreasingorderisconsidered.ItisnotedfromthefigurethatthephasevelocitydecreaseswithincreaseinthemagnitudeofBiot’sgravityparameter G Singhetal.,2017 .Inadditiontothis,thefigureillustratethatthe

impactofBiot’sgravityparameterissignificantatlowfrequencyregionbutlessathighfrequencyregion.Infact,athighfrequencyregion,allthecurvesseemtosharealmostacommonmagnitudeofphasevelocity.6 CONCLUSION

Theeffectsofundulation,corrugation,bondingofthelayerandhalf‐space,initialstressandgravityonthephaseve‐locityofRayleigh‐typewavepropagatinginacorrugatedfibre‐reinforcedlayerlyingoveraninitiallystressedortho‐tropichalf‐spaceundergravityareinvestigated.Theoutcomesofthestudyaresummarizedasfollows:

ThephasevelocityofRayleigh‐typewavedecreaseswithincreaseinwavenumber. ThecorrugationparameterassociatedwiththeupperboundarysurfacefavorsthephasevelocityofRayleigh‐typewavewhereascorrugationparameterassociatedwiththecommoninterfaceoflayerandhalf‐spacedisfa‐vorsthephasevelocityofRayleigh‐typewave.

ThephasevelocityRayleigh‐typewaveincreaseswithincreaseinundulationparameterandpositionparame‐ter.

PhasevelocityofRayleigh‐typewaveincreaseswithincreaseintheinitialstress.Moreprecisely,phasevelocityofRayleigh‐typewavedecreaseswithincreaseinthehorizontaltensileinitialstressbutitincreaseswithin‐creaseinhorizontalcompressiveinitialstress.

Withincreaseinthemagnitudeofbondingparameterofthecommoninterface,thephasevelocityofRayleigh‐typewavedecreases.InparticularphasevelocityofRayleigh‐typewaveismaximumincaseofperfectcontactandleastincaseofsmoothcontactoflayerandhalf‐space

Biot’sgravityparameterdisfavorsthephasevelocityofRayleigh‐typewave. AlthoughtheaffectingparametershavesignificanteffectonthephasevelocityofRayleigh‐typewaveyettheeffectofpresenceandabsenceofcorrugationattheboundarysurfacesonthedispersioncurveisfoundtobegreat.

Thepresentproblemmayfindsomeapplicationsinthefieldofconstruction,civilengineering,geophysicsandgeomechanics.Lowweightandhighstrengthoffibre‐reinforcedmaterialsmakesitacrucialmaterialforvariouscon‐structionworkslikebridges,buildings,towers,etc.Thereinforcementofsoilenhancesthestrengthandloadbearingcapacityofit.Therefore,itisveryimportanttostudytheeffectofdifferentfactorsonthepropagationofwavesthroughthesematerialmediumwithcomplexgeometriesinviewofitspossibleapplicationsindiverseareasofscienceandengineering.

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

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Acknowledgements

TheauthorsconveytheirsincerethankstotheNationalBoardofHigherMathematics NBHM fortheirfinancialsup‐porttocarryoutthisresearchworkthroughProjectNBHM/R.P.78/2015/Fresh/2017/24.1.2017titled“Mathemat‐icalmodelingofelasticwavepropagationinhighlyanisotropicandheterogeneousmedia.”References

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APPENDIX

1 2

11 1 1 2 1 2 12 ,k f HT T Tt f i f e

2 2

12 2 2 2 2 2 22 ,k f HT T Tt f i f e

1 2

13 1 1 2 1 2 12 ,k f HT T Tt f i f e

2 2

14 2 2 2 2 2 22 ,k f HT T Tt f i f e

15 160, 0,t t

1 2

21 1 2 1 1 1 22 4 2 ,k f HT T L Tt f i f e

2 2

22 2 2 2 2 2 22 4 2 ,k f HT T L Tt f i f e

1 2

23 1 2 1 1 1 22 4 2 ,k f HT T L Tt f i f e

2 2

24 2 2 2 2 2 22 4 2 ,k f HT T L Tt f i f e

25 260, 0,t t 1 1'31 1 1 1 1 1 12 ,k f

T T Tt f i f e

2 132 2 2 1 2 2 12 ,k f

T T Tt f i f e 1 1

33 1 1 1 1 1 12 ,k fT T Tt f i f e

2 1'34 2 2 1 2 2 12 ,k f

T T Tt f i f e

1 12 2 ' 2 '35 13 33 1 1 44 1 1 13 1 33 1 1 44 1 1 442 ,iks ft k C C s f C s ik k C s C s f C s f C e

2 12 2 ' 2 '36 13 33 2 1 44 2 2 13 2 33 2 1 44 2 1 442 ,iks ft k C C s f C s ik k C s C s f C s f C e

1 141 1 1 1 1 1 12 4 2 ,k f

T T L Tt f i f e

2 142 2 1 2 2 2 12 4 2 ,k f

T T L Tt f i f e 1 1

43 1 1 1 1 1 12 4 2 ,k fT T L Tt f i f e

2 144 2 1 2 2 2 12 4 2 ,k f

T T L Tt f i f e

112 2 245 44 1 1 11 1 13 1 1 44 1 44 1 11 1 1 13 12 ,

iks ft C ks f C P k f C P ks ik k C s C f C P s f C P s e

12

246 44 2 1 11 1 13 2

2 22 44 2 44 1 11 2 1 13 2

2

,iks f

t C ks f C P k f C P ks

ik k C s C f C P s f C P s e

1 1 2 1 1 1 2 1 1 1 2 12 251 1 52 2 53 1 54 2 55 1 1 56 2 2, , , , , ,k f k f k f k f iks f iks ft e t e t e t e t k k s ki e t k k s ki e

1 1161 1 44 1 1 1 1 12 4 2 ,

1k f

T T L TT

t k ikC c f k ik f e

2 1162 2 44 1 2 2 2 12 4 2 ,

1k f

T T L TT

t k ikC c f k ik f e

1 1163 1 44 1 1 1 1 12 4 2 ,

1k f

T T L TT

t k ikC c f k ik f e

A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave

LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e01 15/15

2 1164 2 44 1 2 2 2 12 4 2 ,

1k f

T T L TT

t k ikC c f k ik f e

1 1 2 165 44 1 1 66 44 2 2, .

1 1iks f iks ft ikC ks k i e t ikC ks k i e