effect of loose bonding and corrugated boundary surface on ...t300/5208 graphite/ epoxy composite by...
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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:[email protected],[email protected],[email protected]*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
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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.
A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave
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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
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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.
A.K.Singhetal./EffectofLooseBondingandCorrugatedBoundarySurfaceonPropagationofRayleigh‐TypeWave
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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