Fundamental solutions in elastodynamics: a compendium

262 Pages·2006·1.793 MB·English

by  Eduardo Kausel

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FUNDAMENTALSOLUTIONSINELASTODYNAMICS
Thisworkisacompilationoffundamentalsolutions(orGreen’sfunctions)for
classicalorcanonicalproblemsinelastodynamics,presentedwithacommon
formatandnotation.Theseformulasdescribethedisplacementsandstresses
elicitedbytransientandharmonicsourcesinsolidelasticmediasuchasfull
spaces,half-spaces,andstrataandplatesinbothtwoandthreedimensions,using
thethreemajorcoordinatesystems(Cartesian,cylindrical,andspherical).Such
formulasareusefulfornumericalmethodsandpracticalapplicationtoprob-
lemsofwavepropagationinelasticity,soildynamics,earthquakeengineering,
mechanicalvibration,orgeophysics.Togetherwiththeplotsoftheresponse
functions,thisworkshouldserveasavaluablereferencetoengineersandsci-
entistsalike.Theseformulaswereheretoforefoundonlyscatteredthroughout
theliterature.Thesolutionsaretabulatedwithoutproof,butgivingreference
toappropriatemodernpapersandbookscontainingfullderivations.Mostfor-
mulasinthebookhavebeenprogrammedandtestedwithintheMATLAB
environment,andtheprogramsthusdevelopedarebothlistedandavailable
forfreedownload.
EduardoKauselearnedhisfirstprofessionaldegreein1967,graduatingasa
civilengineerfromtheUniversityofChile,andthenworkedatChile’sNational
ElectricityCompany.In1969hecarriedoutpostgraduatestudiesattheTech-
nicalUniversityinDarmstadt.HeearnedhisMasterofScience(1972)and
DoctorofScience(1974)degreesfromMIT.Followinggraduation,Dr.Kausel
workedatStoneandWebsterEngineeringCorporationinBoston,andthen
joinedtheMITfacultyin1978,wherehehasremainedsince.Heisaregis-
teredprofessionalengineerintheStateofMassachusetts,isaseniormember
ofvariousprofessionalorganizations(ASCE,SSA,EERI,IACMG),andhas
extensiveexperienceasaconsultingengineer.
Among the honors he has received are a 1989 Japanese Government
Research Award for Foreign Specialists from the Science and Technology
Agency,a1992HonoraryFacultyMembershipinEpsilonChi,the1994Konrad
Zuse Guest Professorship at the University of Hamburg in Germany, the
HumboldtPrizefromtheGermanGovernmentin2000,andthe2001MIT-
CEEAwardforConspicuouslyEffectiveTeaching.
Dr.Kauselisbestknownforhisworkondynamicsoil–structureinteraction,
andforhisverysuccessfulGreen’sfunctions(fundamentalsolutions)forthe
dynamicanalysisoflayeredmedia,whichareincorporatedinanowwidely
usedprogram.Dr.Kauselistheauthorofmorethan150technicalpapers
andreportsintheareasofstructuraldynamics,earthquakeengineering,and
computationalmechanics.
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Fundamental Solutions
in Elastodynamics
A Compendium
EDUARDO KAUSEL
MassachusettsInstituteofTechnology
iii
iii
  
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo
Cambridge University Press
TheEdinburghBuilding,Cambridge,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Information on this title: www.cambridg e.org /9780521855709
©EduardoKausel2006
Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof
relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace
withoutthewrittenpermissionofCambridgeUniversityPress.
Firstpublishedinprintformat
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CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofs
forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot
guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
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Contents
Preface pageix
SECTIONI: PRELIMINARIES
1. Fundamentals 1
1.1 Notationandtableofsymbols 1
1.2 Signconvention 4
1.3 Coordinatesystemsanddifferentialoperators 4
1.3.1 Cartesiancoordinates 4
1.3.2 Cylindricalcoordinates 6
1.3.3 Sphericalcoordinates 9
1.4 Strains,stresses,andtheelasticwaveequation 13
1.4.1 Cartesiancoordinates 13
1.4.2 Cylindricalcoordinates 16
1.4.3 Sphericalcoordinates 21
2. Dipoles 27
2.1 Pointdipolesordoublets:singlecouplesandtensilecrack
sources 27
2.2 Linedipoles 29
2.3 Torsionalpointsources 30
2.4 Seismicmoments(doublecoupleswithnonetresultant) 30
2.5 Blastloads(explosivelineandpointsources) 31
2.6 Dipolesincylindricalcoordinates 32
SECTIONII: FULLSPACEPROBLEMS
3. Two-dimensionalproblemsinfull,homogeneousspaces 35
3.1 Fundamentalidentitiesanddefinitions 35
3.2 Anti-planelineload(SHwaves) 35
3.3 SHlineloadinanorthotropicspace 36
3.4 In-planelineload(SV-Pwaves) 38
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3.5 Dipolesinplanestrain 40
3.6 Lineblastsource:suddenlyappliedpressure 43
3.7 Cylindricalcavitysubjectedtopulsatingpressure 44
4. Three-dimensionalproblemsinfull,homogeneousspaces 48
4.1 Fundamentalidentitiesanddefinitions 48
4.2 Pointload(Stokesproblem) 48
4.3 Tensioncracks 52
4.4 Doublecouples(seismicmoments) 54
4.5 Torsionalpointsource 55
4.6 Torsionalpointsourcewithverticalaxis 57
4.7 Pointblastsource 58
4.8 Sphericalcavitysubjectedtoarbitrarypressure 59
4.9 Spatiallyharmoniclinesource(21/2-Dproblem) 63
SECTIONIII: HALF-SPACEPROBLEMS
5. Two-dimensionalproblemsinhomogeneoushalf-spaces 69
5.1 Half-plane,SHlinesourceandreceiveranywhere 69
5.2 SHlineloadinanorthotropichalf-plane 70
5.3 Half-plane,SV-Psourceandreceiveratsurface(Lamb’s
problem) 71
5.4 Half-plane,SV-Psourceonsurface,receiveratinterior,orvice
versa 73
5.5 Half-plane,lineblastloadappliedinitsinterior(Garvin’s
problem) 76
6. Three-dimensionalproblemsinhomogeneoushalf-spaces 78
6.1 3-Dhalf-space,suddenlyappliedverticalpointloadonits
surface(Pekeris-Mooney’sproblem) 78
6.2 3-Dhalf-space,suddenlyappliedhorizontalpointloadonits
surface(Chao’sproblem) 81
6.3 3-Dhalf-space,buriedtorsionalpointsourcewithverticalaxis 83
SECTIONIV: PLATESANDSTRATA
7. Two-dimensionalproblemsinhomogeneousplatesandstrata 87
7.1 PlatesubjectedtoSHlinesource 87
7.1.1 Solutionusingthemethodofimages 87
7.1.2 Normalmodesolution 87
7.2 StratumsubjectedtoSHlinesource 88
7.2.1 Solutionusingthemethodofimages 89
7.2.2 Normalmodesolution 90
7.3 PlatewithmixedboundaryconditionssubjectedtoSV-Pline
source 90
7.3.1 Solutionusingthemethodofimages 91
7.3.2 Normalmodesolution 92
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Contents vii
SECTIONV: ANALYTICALANDNUMERICALMETHODS
Readmefirst 97
8. SolutiontotheHelmholtzandwaveequations 98
8.1 Summaryofresults 98
8.2 ScalarHelmholtzequationinCartesiancoordinates 101
8.3 VectorHelmholtzequationinCartesiancoordinates 102
8.4 ElasticwaveequationinCartesiancoordinates 104
8.4.1 Horizontallystratifiedmedia,planestrain 105
8.5 ScalarHelmholtzequationincylindricalcoordinates 108
8.6 VectorHelmholtzequationincylindricalcoordinates 109
8.7 Elasticwaveequationincylindricalcoordinates 110
8.7.1 Horizontallystratifiedmedia 112
8.7.2 Radiallystratifiedmedia 114
8.8 ScalarHelmholtzequationinsphericalcoordinates 117
8.9 VectorHelmholtzequationinsphericalcoordinates 118
8.10 Elasticwaveequationinsphericalcoordinates 120
9. Integraltransformmethod 125
9.1 Cartesiancoordinates 125
9.2 Cylindricalcoordinates 130
9.2.1 Horizontallystratifiedmedia 130
9.2.2 Cylindricallystratifiedmedia 136
9.3 Sphericalcoordinates 137
10. Stiffnessmatrixmethodforlayeredmedia 140
10.1 Summaryofmethod 141
10.2 StiffnessmatrixmethodinCartesiancoordinates 142
10.2.1 Analyticcontinuationinthelayers 148
10.2.2 Numericalcomputationofstiffnessmatrices 150
10.2.3 Summaryofcomputation 151
10.3 Stiffnessmatrixmethodincylindricalcoordinates 159
10.3.1 Horizontallylayeredsystem 160
10.3.2 Radiallylayeredsystem 164
10.4 Stiffnessmatrixmethodforlayeredspheres 175
10.4.1 Propertiesanduseofimpedancematrices 180
10.4.2 Asymmetry 180
10.4.3 Expansionofsourceanddisplacementsintospherical
harmonics 181
10.4.4 Rigidbodyspheroidalmodes 182
SECTIONVI: APPENDICES
11. Basicpropertiesofmathematicalfunctions 185
11.1 Besselfunctions 185
11.1.1 Differentialequation 185
11.1.2 Recurrencerelations 185
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viii Contents
11.1.3 Derivatives 186
11.1.4 Wronskians 186
11.1.5 Orthogonalityconditions 186
11.1.6 Usefulintegrals 187
11.2 SphericalBesselfunctions 187
11.2.1 Differentialequation 187
11.2.2 Trigonometricrepresentations 188
11.2.3 Recurrencerelations 189
11.3 Legendrepolynomials 189
11.3.1 Differentialequation 189
11.3.2 Rodrigues’sformula 190
11.3.3 Trigonometricexpansion(x=cosφ) 190
11.3.4 Recurrencerelations 190
11.3.5 Orthogonalitycondition 190
11.3.6 ExpansioninLegendreseries 191
11.4 AssociatedLegendrefunctions(spheroidalharmonics) 191
11.4.1 Differentialequation 191
11.4.2 Recurrencerelations 192
11.4.3 Orthogonalityconditions 192
11.4.4 Orthogonalityofco-latitudematrix 193
11.4.5 Expansionofarbitraryfunctioninspheroidal
harmonics 193
11.4.6 Leibnizruleforthederivativeofaproductoftwo
functions 193
12. Brieflistingofintegraltransforms 194
12.1 Fouriertransforms 194
12.2 Hankeltransforms 196
12.3 SphericalHankeltransforms 197
13. MATLABprograms 198
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Preface
Wepresentinthisworkacollectionoffundamentalsolutions,orso-calledGreen’sfunc-
tions,forsomeclassicalorcanonicalproblemsinelastodynamics.Suchformulasprovide
thedynamicresponsefunctionsfortransientpointsourcesactingwithinisotropic,elastic
media,inboththefrequencydomainandthetimedomain,andinbothtwoandthree
dimensions.Thebodiesconsideredarefullspaces,half-spaces,andplatesofinfinitelateral
extent,whilethesourcesrangefrompointandlineforcestotorques,seismicmoments,and
pressurepulses.Byappropriateconvolutions,thesesolutionscanbeextendedtospatially
distributedsourcesand/orsourceswithanarbitraryvariationintime.
Thesefundamentalsolutions,astheirnameimplies,constituteinvaluabletoolsfora
largeclassofnumericalsolutiontechniquesforwavepropagationproblemsinelasticity,
soildynamics,earthquakeengineering,orgeophysics.ExamplesaretheBoundaryIntegral
(orelement)Method(BIM),whichisoftenusedtoobtainthesolutiontowavepropagation
problemsinfinitebodiesofirregularshape,evenwhileworkingwiththeGreen’sfunctions
forafullspace.
Thesolutionsincludedhereinarefoundscatteredthroughouttheliterature,andno
singlebookwasfoundtodealwiththemallinoneplace.Inaddition,eachauthor,paper,
orbookusessignconventionsandsymbolsthatdifferfromoneanother,ortheyinclude
onlypartialresults,sayonlythesolutioninthefrequencydomainorforsomeparticular
valueofPoisson’sratio.Sometimes,publishedresultsarealsodisplayedinunconventional
manners,forexample,takingforcestobepositivedown,butdisplacementsup,orscaling
thedisplaysinunusualwaysorusingtoosmallascale,andsoforth.Thus,itwasfelt
thatacompendiumoftheknownsolutionsinacommonformatwouldserveauseful
purpose.Withthisinmind,weusethroughoutaconsistentnotation,coordinatesystems,
andsignconvention,whichshouldgreatlyfacilitatetheapplicationofthesefundamental
solutions.Also,whileweanguishedinitiallyatthechoiceofsymbolsfortheanglesin
sphericalcoordinates,wedecidedintheendtouseθfortheazimuthandφforthepolar
angle.Althoughthiscontravenesthecommonnotation,itprovidesconsistencybetween
sphericalandcylindricalcoordinatesandeasesthetransitionbetweenoneandtheother
system.
Wetabulatethesesolutionshereinwithoutproof,givingreferencetoappropriatemod-
ernpapersandbookscontainingfullderivations,butmakingnoeffortatestablishingthe
originalsourcesofthederivationsor,forthatmatter,providingahistoricalaccountof
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