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Publié par | universitat_stuttgart |
Publié le | 01 janvier 2005 |
Nombre de lectures | 22 |
Poids de l'ouvrage | 1 Mo |
Extrait
Regularityresultsforquasilinearellipticsystems
ofpower-lawgrowthinnonsmoothdomains
—Boundary,transmissionandcrackproblems—
VonderFakult¨atMathematikundPhysikderUniversit¨atStuttgart
zurErlangungderW¨urdeeines
DoktorsderNaturwissenschaften(Dr.rer.nat.)
Abhandlunggenehmigte
onvorgelegtv
KneesDorotheeDipl.-Math.
MitbHauptberichericter:hter:
geboreninLeonberg
andigS¨A.-M.Dr.Prof.rehseFJ.Dr.Prof.A.Dr.Prof.eMielk
Tagderm¨undlichenPr¨ufung:30.11.2004
Institutf¨urAngewandteAnalysisundNumerischeSimulation
2005
Dorothee
Knees
Fachbereich
Institut
ur¨f
atersit¨Univ
Mathematik
andteAngew
Stuttgart
aldringwPfaffen
D-70
569
57
Stuttgart
Analysis
knees@ians.uni-stuttgart.de
und
heNumerisc
ulationSim
Vortorw
DievorliegendeDissertationentstandimRahmendesSonderforschungsbereichs404
”MehrfeldproblemeinderKontinuumsmechanik“,TeilprojektC5”Spannungssingu-
larit¨ateninheterogenenMaterialien“anderUniversit¨atStuttgart.
AndieserStellem¨ochteichmichsehrherzlichbeiFrauProf.Dr.A.-M.S¨andig
f¨urdieAufgabenstellung,dieumfassendeBetreuungunddieF¨orderung,diesiemir
zuteilwerdenließ,bedanken.SiehattejederzeiteinoffenesOhrf¨urmeineFragen.
HerrnProf.Dr.J.FrehseundHerrnProf.Dr.A.Mielkedankeichf¨urdie¨Ubernahme
desKoreferatsundihrekonstruktiveKritikanmeinerArbeit.
Dar¨uberhinausm¨ochteichmichbeiallenAngeh¨origendes6.LehrstuhlsdesIn-
stitutsf¨urAngewandteAnalysisundNumerischeSimulationundinsbesonderebei
HerrnProf.Dr.-Ing.W.L.Wendlandf¨urdieangenehmeArbeitsatmosph¨areunddie
zahlreicheninspirierendenDiskussionenimOberseminarundw¨ahrendderKaffee-
en.edankbpausen
EinbesondererDankgehtanmeineElternundBr¨uder,diemichdieganzeZeit
mitvielLiebeunterst¨utzten.
JanimStuttgart,2005uar
iii
KneesDorothee
tstenCon
Preface(German)
tstenCon
ductiontroIn
Zusammenfassung
1Physicalandmathematicalbasics
1.1DescriptionoftheRamberg/Osgoodmodel..............
1.1.1Strainsandstresses.......................
1.1.2TheRamberg/Osgoodrelation.................
1.1.3Theboundarytransmissionproblem.............
1.1.4Planestates...........................
1.2Domainsandfunctionspaces......................
1.2.1Admissibledomains.......................
1.2.2Notationformatrices......................
1.2.3Functionspaces.........................
1.3WeakformulationoftheRamberg/Osgoodequations........
1.3.1Weakformulationandexistenceofsolutions.........
1.3.2Complementaryenergyandelasticstrainenergy.......
1.4Quasilinearellipticsystemsofp-structure...............
2Localregularityandregularityatsmoothpartsoftheboundary
2.1Systemsofp-structure.........................
2.2Ramberg/Osgoodequation.......................
2.2.1Auxiliarylemmata.......................
2.2.2Proofofregularitytheorem2.3................
2.3Regularityatplanepartsofboundariesandinterfaces........
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1313141518182020212130303539
434345474955
vi
tstenCon
3Globalregularityfortransmissionproblems61
3.1Motivation................................62
3.1.1Poissonequationwithpiecewiseconstantcoefficients....62
p,13.1.2DifferencequotientsandthespaceW(Ω)..........64
3.2Quasi-monotonecoveringconditionfortransmissionproblems...65
3.2.1Definitionforballs.......................65
3.2.2DefinitionforLipschitziandomains..............70
3.3Globalregularityforequationsofp-structure.............74
3.3.1Maintheoremfortransmissionproblemsofp-structure...74
3.3.2Specialcase:Linearellipticproblems,d=2.........76
3.3.3Proofoftheorem3.18......................78
3.4GlobalregularityforRamberg/Osgoodequations..........90
3.4.1Thequasi-monotonecoveringconditionforRamberg-
Osgoodequations........................91
3.4.2Regularityresultsandspecialcases..............92
3.4.3HRR-fieldscomparedwiththeregularityresults.......93
3.4.4Proofoftheorem3.29......................96
3.4.5Improvedregularityatplanepartsoftheboundary.....103
4GlobalregularityofthestressesintheHenckymodel107
4.1TheHenckymodel...........................107
4.2Globalregularityofthestressfield..................109
4.3Proofoftheorem4.2..........................110
5FracturemechanicsfortheRamberg/Osgoodmodel119
5.1Griffith’senergycriterion........................121
5.1.1Energies.............................121
5.1.2Admissibleneighbouringconfigurations............122
5.1.3Energyreleaserate.......................124
5.2Griffith’sformulaandJ-integral....................125
5.2.1Linearelasticmaterials.....................125
5.2.2Ramberg/Osgoodmodel....................126
5.2.3Proofoftheorem5.5......................128
6Summaryoftheregularityresults
146
149inequalitiestialEssenAA.1Generalinequalities...........................149
A.2Inequalitiesforconjugatefunctions..................151
A.3InequalitiesfortheRamberg/Osgoodmodel.............153
A.3.1Thegeneralcase........................153
tenContsA.3.2Planestress................
A.3.3Aprioriestimates.............
BMoreaboutfunctionspaces
,sq,sqB.1ThespacesΣ(Ω)andΣ(Ω).........
p,rB.2ThespaceU(Ω).................
B.3ThespaceLD(Ω).................
theoremsurtherFCC.1Neˇcas’lemma...................
C.2Meanvaluetheorem...............
C.3Aminimisationproblemwithaffineconstraints
yBibliograph
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Thethesisisdevotedtotheanalysisofdisplacementandstressfieldsofheteroge-
neouselasticbodieswhicharesubjectedtoexternalloadings.Theheterogeneous
bodiesarecomposedofseveralnonlinearelasticsubstructureswhichobeyacon-
stitutiverelationofpower-lawtype.Anexampleforsuchaconstitutiverelation
wasintroducedbyW.RambergandW.R.Osgoodin[80,1943],(Ramberg/Osgood
model,alsoknownasNorton/Hoffmodel).Thewholebodyaswellastheinter-
nalinterfaces,whichseparatesubstructureswithdifferentmaterialproperties,may
havecornersandedges.Physicalexperimentsaswellasnumericalcomputations
showthatveryhighstressconcentrationscanoccurinthevicinityofre-entrant
corners,cracks,edgesandnearthosepointswherethematerialparametersaredis-
continuous.Thesestressconcentrationshaveastronginfluenceonthestrengthand
physicallifeofthebody.Inparticularcrackscandevelopduetothesestresssingu-
laritiesandcanleadtothefailureofthewholestructure.Manystrengthhypotheses
infracturemechanicsarebased(moreorlessexplicitly)onthestressdistribution
inthebodyandthereforeagoodknowledgeofthestressesisimportant.More-
over,convergenceratesofstandardnumericalschemesdecreaseatthepresenceof
stresssingularities.Here,thea-prioriknowledgeofpossiblestressdistributionscan
beusedtodevelopimprovedalgorithmswheree.g.specialsingularfunctionsare
includedintheFE-spacesorwherethemeshesaresuitablyrefinedinadvancein
regions,wherehighstressesmayoccur.
Amathematicalformulationofthefieldequationsoftheseheterogeneousbodies
leadstosystemsofquasilinearellipticpartialdifferentialequationswithpiecewise
constantcoefficients.Thesesystemsarecloselyrelatedtothemoregeneralclass
ofquasilinearellipticpartialdifferentialequationsofp-structure,forwhichthep-
Laplaceequationisatypicalexample.Motivatedbytheseobservationsthemain
goalsofthisthesisarethefollowing:
•
ationDerivofglobalyregularitresultsfortransmissionproblemsforsystemsofquasilinearellipticpartialdifferentialequationsofp-structureonnonsmooth
domains.
1
2
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•Derivationofglobalregularityresultsfordisplacementandstressfieldsof
heterogeneousbodieswhicharecomposedofdifferentphysicallynonlinear
elasticmaterialsofRamberg/Osgood-type.
Weemphasisethatdifferentgrowthpropertiesofthedifferentialoperatorsonneigh-
bouredsubdomainsareadmitted.Themathematicalpropertiesofquasilinearequa-
tionsofp-structureandofthefieldequationsoftheRamberg/Osgoodmodelare
closelyrelatedandthereforesimilarbasicideasandtechniquescanbeappliedfor
thederivationoftheregularityresults.Howeverbothclassesofequationsneeda
differenttreatmentindetail.
Twoapplicationsoftheregularityresultsarepresented.Wederiveaglobalregu-
smolarityothresultdomainsfortheusingstressthefieldsresultsofforHencstressky’sfieldselastic-poftheerfectlyRambplasticerg/Osgomodelodformonon-del.
TheproofisbasedonatheorembyR.Temam[100]andA.Bensoussan/J.Frehse[5]
bystatingstressthatfieldsstresswhichfieldsarewhichsolutionsareofsolutionstheRamofbHencerg/Osgoky’somoddelcanequationsbewithapprosuitablyximated
chosenmaterialparameters.AssecondapplicationwestudyGriffith’sfracturecri-
terionforRamberg/Osgoodmaterials.Griffith’sfracturecriterionisanenergetic
criterionandcanbereformulatedintermsoftheenergyreleaserate.Variousfor-
mulasforthecalculationoftheenergyreleaseratearepresentedintheengineering
literature.Butfromamathematicalpointofviewitisa-priorinotobviouswhether
theexpressionsintheseformulasarewelldefinedforweaksolutionsoftheRam-
boftheseerg/Osgoformodulas.equations.Summa