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International Journal of Mechanical Sciences
journalhomepage:www.elsevier.com/locate/ijmecsci
Fluid-structure interaction effects during the dynamic response of clamped thin steel plates exposed to blast loading
Vegard Aune
a,b,∗, Georgios Valsamos
c, Folco Casadei
c, Magnus Langseth
a,b, Tore Børvik
a,baStructural Impact Laboratory (SIMLab), Department of Structural Engineering, NTNU - Norwegian University of Science and Technology, Trondheim, Norway
bCentre for Advanced Structural Analysis (CASA), NTNU, Trondheim, Norway
cEuropean Commission, Joint Research Centre (JRC), Ispra (VA), Italy
a r t i c le i n f o
Keywords:
Lightweight structures Blast mitigation Shock tube Numerical simulations EUROPLEXUS
a b s t r a ct
Thisworkpresentsresultsfromanumericalinvestigationontheinfluenceoffluid-structureinteraction(FSI)on thedynamicresponseofthinsteelplatessubjectedtoblastloading.Theloadingwasgeneratedbyashocktube testfacilitydesignedtoexposestructurestoblast-likeloadingconditions.Thesteelplateshadanexposedareaof 0.3m×0.3mandexperiencedlargedeformationsduringthetests.Numericalsimulationswereperformedusing thefiniteelementcodeEUROPLEXUS.AnuncoupledFSIapproachwascomparedtoacoupledFSIapproachin anattempttoinvestigateFSIeffects.Reduceddeformationwasobservedintheplatesduetotheoccurrenceof FSIduringthedynamicresponse.ThegeneraltrendwasanincreasedFSIeffectwithincreasingblastintensity.
Thenumericalresultswerefinallycomparedtotheexperimentaldatatovalidatetheirreliabilityintermsof deflectionsandvelocitiesinthesteelplates.Agoodagreementwiththeexperimentaldatawasfound,andthe numericalsimulationswereabletopredictboththedynamicresponseoftheplateandthepressuredistributionin frontoftheplatewithgoodaccuracy.Hence,thenumericalframeworkpresentedhereincouldbeusedtoobtain moreinsightregardingtheunderlyingphysicsobservedintheexperiments.Theclearconclusionfromthisstudy isthatFSIcanbeutilizedtomitigatetheblastloadactingonaflexible,ductileplatedstructure,resultingin reduceddeformations.
1. Introduction
Civilengineering structuresextend thescope of traditionalblast- resistantdesignbyalsoincludingarchitectural,lightweightandflexi- blestructures[1–4].Thesetypesofstructuresmayexperiencesevere blast-structureinteractionbetweenthepropagatingblastwaveandthe structuralresponse(see,e.g.,[5–9]).Tomeetthechallengesposedby suchextremeloadingconditions,itisnecessarytofullyunderstandthe importanceoftheseinteractionsinviewofblast-resistantdesign.Blast- structureinteractionoccurswhentheblastwaveencountersastructural surfacethatisnotparalleltothedirectionofthewave.Theblastwave isthenreflectedandreinforced.Dependingontheblastandstructural properties,thestructuretypicallybehavesaseitherarigidordeformable surface.Fluid-structureinteraction(FSI)takesplaceifthestructuralsur- faceisallowedtomoveordeform.
Taylor[10]isconsideredtobeoneofthepioneersinthefieldof FSIinblastenvironments,suggestingthatlightweightstructuresunder- takelessmomentumcomparedtoheavierstructureswhenexposedto
∗Correspondingauthorat:StructuralImpactLaboratory(SIMLab),DepartmentofStructuralEngineering,NTNU-NorwegianUniversityofScienceandTechnology, Trondheim,Norway.
E-mailaddress:vegard.aune@ntnu.no(V.Aune).
thesameblastintensity.Thatis,themotionofthereflectingsurface reducesthepressureactingonit.Recentyearshaveseenasignificant increaseintheamountofresearchinvestigatingtheinfluenceofFSIef- fectsontheresponseofblast-loadedplates.Mostofthesestudieshave focusedonplatedstructuresinunderwaterblastenvironments[11–15]. Theseinvestigationstypicallyassumedanacousticmediumcharacter- izedbyanincompressiblefluidandlinearsuperpositionofweakshock waves.Althoughtheneedtoaccountforacompressiblefluidbehaviour was recognized[8,12,16,17], thiswasnot takenintoaccountduring FSIinairblastenvironmentsuntiltheworksofKambouchevetal.[6], Kambouchev[18],Kambouchevetal.[19,20],VaziriandHutchinson [21]andHutchinson[22].Theacousticassumptionholdsforunderwa- terexplosions,butcompressibilityeffectsaresignificantinairevenfor smallmagnitudesofblastoverpressures.Thecompressiblebehaviourof airresultsinasignificantincreaseinthemagnitudeofthestagnation pressureexperiencedbythestructureduringtheblast-structureinterac- tionsincethereflectedoverpressureincreaseswiththeincidentpressure inahighlynon-linearmanner.Abasicunderstandingoftheinfluence
https://doi.org/10.1016/j.ijmecsci.2020.106263
Received12August2020;Receivedinrevisedform17December2020;Accepted27December2020 Availableonline4January2021
0020-7403/© 2021TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)
ofFSIwhentheblastwave(inacompressiblefluid)interactswitha movableordeformablesurfaceisgivenintheworksofCourantand Friedrichs[23],Toro[24]andSubramaniametal.[25].Ifthestructure startstomove,themotionaltersthepressureatits surface.Previous researchhasshownthatFSIeffectscanmitigatetheblastloadacting onthestructure[19–21,26],especiallyinsituationsinvolvinglargede- formations[6,7,25,27].Theblastmitigationhasbeenrelatedtoboth theinducedvelocity[7,25]andtothedeformed shapeof thestruc- ture[26,28,29].Thisisinterestinginviewoflightweightandflexible structures.Lightweightstructureswillexperienceahigherinducedve- locityandareductionin thetransmittedimpulseafterimpactofthe blastwave,whileflexiblestructureswillexperiencelargeinelasticde- formation(seeRef.[30,31])andapossibleinteractionofthedynamic responsewiththepositivephaseoftheload.Thisimpliesthatlargede- formationsandenergyabsorptioninstructuralmembersarefavourable sincetheblastwaveismitigatedthroughvariousdeformationmecha- nismsinthestructure.Aslongasthestructuralmembercansustainthe deformationthatariseswithoutexperiencingfailure,ductilematerials canbeutilizedinthedesignofflexiblestructuresbyallowingforfinite deformations.TheFSImaythenreducethetransmittedimpulseandin- creasetheblastperformanceofthestructure.However,exploitingthis mitigationeffectintheblast-resistantdesignrequiresathoroughunder- standingofthegoverningphysicsintheproblem.
Althoughapproximatemethodsmayprovidedesignguidance,these methodsareoftenbasedonseveralassumptionsregardingthespatial andtemporal distributionof the loading. Advanced numerical tech- niquesarethereforeoftenrequiredforasufficientinsightinboththe loadingandtheresultingdynamicresponse.Awidelyuseddesigntool forthisclassofproblemsistheexplicitnon-linearfiniteelement(FE) method[32].Theuncoupledapproachisoftenthepreferredprocedure intoday’sblast-resistantdesign.Theloadingisthenobtainedusingei- therempiricalrelationsfromtheliteratureornumericalsimulationsof theblastwavepropagationinanEulerian(fixed)referenceframe.The underlyingassumption in thisapproachis rigidboundaryconditions andnodeformationofthestructure,wherethenumericalsimulations aretypicallyperformedinacomputationalfluiddynamics(CFD)code.
These typesof codes computethefluidflow andprovidethespatial andtemporalpressuredistributionalongthefluidboundary.Then,the obtainedpressurehistoryisappliedinacomputationalstructuraldy- namics(CSD)codetodeterminethecorrespondingdynamicresponseof thestructure.Theuncoupledapproachthereforemakestheinherentas- sumptionthattheblastpropertiesareunalteredbythestructuralmotion andviceversa.Sincethebehaviourofblast-loadedsteelplatesishighly non-linear(bothinthegeometryandinthematerial),thismaynotbe anadequateapproachandcouldresultinanon-physicalresponse.Both thepressuredistributionandthedynamicresponsecanbesignificantly influencedbyFSIeffects.ThiswasillustratedbyCasadeietal.[5]and Børviketal.[7]bycomparinguncoupledandfullycoupledFSIsimula- tionsfortypicalindustrialapplications.Børviketal.[7]observedcon- siderablevariationsinthepredictedresultsfromuncoupledandcoupled methodsandemphasizedtheimportanceofanaccuratequantification oftheloading.Recentadvancements[33,34]inthefieldofFEmethods makeitnowpossibletostudytheFSIeffectsinblasteventsinvolving complexgeometries,largedeformations,failureandfragmentation.In particular,adaptivemeshrefinement(AMR)[35–38]inboththefluid (F)andstructural(S)sub-domainsallowsforasufficientlyfinemesh sizetorepresentthenearinstantaneousriseinpressureacrosstheshock waveandtopredictthepressuredistributionsattheF-Sinterface.Nu- mericalsimulationscanthereforebe usedtoinvestigate theeffectof FSIonthedynamicresponseofplatedstructures.However,beforesuch methodscanbe used,itisessential toevaluatetheirperformancein termsofrobustness,reliabilityandeffectivenessinpredictingboththe loadingandthedynamicresponse.Experimentalvalidationisoftenpre- ferredasitrepresentstheactualphysicsintheproblem,andcontrolled experimentsinlaboratoryenvironmentscanbeusedtoevaluatecurrent computationalmethods.
ThismotivatesdetailedinvestigationsonFSIeffectsduringthedy- namic responseof blast-loadedsteel plates.Previous studies[36,39–
41]werenotabletofullyaddressFSIeffectsduringthedynamicre- sponseoftheplates,mainlybecausetheloadingwassignificantlyover- estimatedinthenumericalsimulationsatincreasingmagnitudesofpres- sure.Thecurrentworkhasmanagedtoconsiderablyimprovethepredic- tivecapabilitiesofthesimulations,allowingfordetailedstudiesonthe underlyingphysicsduringFSI.Therefore,theobjectivesofthisstudyare asfollows:(1)establishareliablenumericalmethodologybasedonre- centdevelopmentsinEPX;(2)numericallyquantifytheinfluenceofFSI effectsonthedynamicresponseofthinsteelplates;and(3)useexisting experimentaldata[36]toevaluatetheperformanceofthenumerical simulationsandensurethattheunderlyingphysicsarecaptured.
2. Experimentalwork
Experiments were performed in the SIMLab ShockTube Facility (SSTF)atNTNU.Adetailedpresentationofthedesign,evaluationofits performanceandtheexperimentalprogrammeusedhereincanbefound inRefs.[36,39].However,theexperimentalsetupandprogrammeare brieflyrepeatedinthefollowingforcompletenesssincemostofthese testsservedasthebasisforthefinalevaluationofthenumericalsim- ulationsthatwillbepresentedinSection4.TheSSTFhasbeenproven toproducecontrolledandrepeatableblastloadinginlaboratoryenvi- ronments[39],anditisconsideredtobewellsuitedtostudyFSIeffects duringthedynamicresponseofblast-loadedplates(see,e.g.,[36,39–
41]).
Theoverallprincipleisthatofacompressed-gas-drivenshocktube, inwhichahigh-pressurechamber(calleddriverinFig.1a)isseparated fromalow-pressurechamber(calleddriveninFig.1a)byusingmulti- plediaphragms.Asuddenopeningofthediaphragmsgeneratesashock wavetravellingdownthetubeandintothelow-pressurechamber.By usingarelativelysmallratiobetweenthelengthsofthehigh-pressure andlow-pressurechambers,thisexperimentalsetupdiffersfromtradi- tionalshocktubesinthewaythatthereflectedrarefactionwavescatch upwiththeshockwaveresultinginpressureprofilessimilartotheblast wavefromanexplosivedetonation[39,42].
Thetotallengthofthetubeis18.355manditismadefromstain- lesssteelofgradeP355NH,whichisintendedforpressurepurposesac- cordingtotheEN13445.Thehigh-pressurechamber(calleddriverin Fig.1a)ismanufacturedwithatotallengthof2.02mandhasacircular cross-sectionwithaninnerdiameterof0.331m,wheretheinternalwall isdullpolishedtoobtainasmoothsurface.Aluminiuminsertsmaybe usedtoreducetheeffectivelengthofthedriversectionin0.25mincre- ments.Thedriverisfollowedbya0.14-m-longfiringsectionthatcon- sistsofseveralintermediatepressurechambersseparatedbydiaphragms (Fig.1aand1c).Thisenablesthetotalpressuredifferencebetweenthe driveranddrivensectiontobeachievedinastepwisemanner.Thetest startsbyfillingthedriverandfiringsectionwithcompressedair,where thepressuredifferencesintheintermediatechambersareoperatedbe- lowthediaphragmrupturestrengthsuchthatthedesiredpressureisob- tainedinthedriver.Ruptureofthediaphragmsisinitiatedbycontrolled andrapidventingoftheintermediatepressureclosesttothedriversec- tionusingtwosolenoidvalves.Thisensuresacontrolledruptureofthe diaphragmsandreproducibleburstingpressures.Theburstingpressure maybevariedbychangingthethicknessof thediaphragms.Melinex sheetsareusedasdiaphragmsduetothismaterial’sstrength andre- peatability.
Theinnercross-section in thedrivensectionstarts witha 0.6-m- longtransitionregionfromcirculartosquarecross-section(atconstant area),wherethesquarecross-sectioncontinuesuntiltheveryendofthe tube (Fig.1a).Anepoxy materialisusedtoobtainasmoothsurface andasquarecross-sectionof0.3m×0.3minsidethesurroundingtube (Fig.1d).Theepoxymaterialworksasapracticallyincompressiblema- terial,whilethesurroundingtubeensuresthestructuralstrength.The
Fig. 1. Experimental setup of the SIMLab ShockTubeFacility(SSTF):(a)Sketchofthe experimentalsetup(seenfromabove),picture ofthe(b)shocktube(seenfromthedriversec- tion),(c)firingsection(seenfromthedriven section),(d)internalcross-sectionofthedriven section(seenfromthecameras)and(e)clamp- ing andDICspecklepattern fortheflexible steelplate(seenfromthe cameras).Reprint fromAuneetal.[36,39],Aune[41].
Table1
Testmatrixincludinginitialconditionsandrepresentativeblastpropertiesforeachtest.
Test
Initial conditions Blast properties ∗
Pressure (driver) Pressure (driven) Temperature 𝑀 s 𝑝 r,max 𝑡 d+ 𝑖 r+
[kPa] [kPa] [ ◦C] [ - ] [kPa] [ms] [kPa ms]
D05 637.6 100.5 21.9 1.37 267.5 28.7 2557.9
D15 1716.0 100.8 21.4 1.63 606.6 44.1 7510.0
D25 2811.0 100.8 21.0 1.75 795.2 68.7 12,383.3
D35 3914.0 100.7 22.2 1.88 1105.2 73.9 16,613.4
D60 6307.0 100.6 23.0 2.04 1446.1 75.3 21,151.7
∗Representativeblastpropertiesobtainedfrommassive,non-deformableplatesinRef.[39]fromtests withsimilarinitialconditions.
averageroughness(Ra)ofthesurfacesinsidethedrivensectionisre- portedbythemanufacturertobeintherangeof0.2–0.4μm.
Inthepresentwork,thelengthofthedriveranddrivensectionswas 0.77mand16.20m(Fig.1a),respectively,bothwithacross-sectional areaof0.09m2.Theblastintensitywasvariedbychangingtheinitial pressureinthedriversection,whiletheinitialpressureinthedriven sectionwasatambientconditions.Table1givesthetestmatrixused herein,whereeachtestisnumberedDYinwhichDdenotesdeformable steelplate(D)andYindicatesthefiring(absolute)pressureinbarsin thedriver.FromTable1,itisnotedthatthetestnumbersarerounded tothelowermultipleof5forthefiringoverpressures(inbars).Athin Docol600DLsteelplatewasmountedattheendofthetubetointroduce movingboundaryconditions(Fig.1e).Thedeformablesteelplateswith dimensions0.625m×0.625m×0.0008mwereclampedtotheend flangeofthetubeinanattempttoachievefixedboundaryconditions (Fig.1e).Each ofthe12 boltswas tightenedusinga wrenchwitha
torque 𝑀𝑡of200Nm.Thisisequivalenttoapre-tensioningforce𝐹𝑝 of46.6kNfortheM24boltsusedintheSSTF[36].Theplateshadan exposedareaof0.3m×0.3m(equaltotheinternalcross-sectionofthe tube).
To establish a basis for comparison of the dynamic response in thenumericalsimulations,thesteelplateswerespray-paintedwitha specklepattern(Fig.1e)andthree-dimensionaldigitalimagecorrela- tion(3D-DIC)analyseswerecarriedout tomeasurethetransientdis- placementfield.Thestereovisionsetupofthetwohigh-speedcameras (Phantomv2511)isillustratedinFig.1a.The3D-DICwasperformed usingthein-houseDICcodeeCorr[43]comparingthegreyscale-value fieldofthespecklepatternforanimageinthedeformed(current)con- figurationtothatintheundeformed(reference)configuration.
Piezoelectricpressuresensors(Kistler603B)wereusedtomeasure thepressure24.5cm(Sensor1)and34.5cm(Sensor2)upstreamofthe testspecimen(Fig.1a).Thepressuresensorswereflushmountedinthe
Fig.2. WeakcouplingusingCCFVsintheem- beddedFSIapproach:(a)facesintheinfluence domain,(b)calculationof thepressuredrop force𝐟Δ𝑝=(𝑝1−𝑝2)𝐿𝐧𝑓 and(c)improvingal- gorithmspatialresolutionbyFSI-drivenAMR inthefluid(onlyonerefinementlevelshown forsimplicity).(a)and(b)arereprintsfrom Casadeietal.[34].
roofoftheshocktube,automaticallytriggeredwhentheshockwave arrivedatSensor2andoperatedwithasamplingfrequencyof500kHz.
Thepressuremeasurementswerealsosynchronizedwiththehigh-speed camerasoperatingatarecordingrateof24kHz.Thedynamicresponse intermsofmid-pointdeflectionsandthepressuremeasurementsatSen- sor1intestsD05toD35werealreadyreportedinRef.[36].Theexperi- mentalresultswillbepresentedandusedforvalidationofthenumerical simulationsinSection4.
Theblastintensityistypicallyrepresentedasapressurehistory𝑝(𝑡) describedbythepeakreflectedoverpressure𝑝r,max,thedurationofthe positivephase𝑡d+andthepositivespecificimpulse𝑖r+.TheMachnum- ber𝑀sisalsofrequentlyusedtoindicatetheblastintensity.Therepre- sentativeblastpropertiesobtainedfrommassive,non-deformableplates inRef.[39]fromtestswithsimilarinitialconditionsarealsoincluded inTable1forcompleteness.
3. Numericalstudy
ThenumericalsimulationswereperformedbytheexplicitFEcode EUROPLEXUS(EPX) [44],which is jointly developed bythe French Commissariatà l’EnergieAtomiqueetauxEnergiesAlternatives(CEA) andbytheJointResearchCentreoftheEuropeanCommission(JRC).
TheCast3Msoftware[45],alsodevelopedbyCEA,wasusedtogener- atetheFEmeshesforthevariousnumericalmodels,whiletheParaView software[46]andEPXitselfwereusedforpost-processingofthenumer- icalresults.InthefullycoupledFSIsimulationspresentedinthisstudy, boththefluidsub-domainandthestructuralsub-domainareincluded tobeabletostudytheinfluenceofFSIeffectsonthedynamicresponse of blast-loadedsteelplates. Adetailedpresentationof thegoverning equationsforthestructuralandfluidsub-domainscanbefoundinthe Appendix.
3.1. ModellingofFSI
Thefluidsub-domainwasdiscretizedwithcell-centred finitevol- umes(CCFV) becausethey aresuperiortotraditionalfiniteelements (FE)regardingmodellingofdiscontinuitiesinthefluidflow.Coupling betweenthestructuralsub-domainandthefluidsub-domainisachieved byanFSIalgorithmoftheembedded(orimmersed) type,knownas FLSWinEPX;seeRef.[34]andFig.2.Thisparticularalgorithmischo- senamongvariousotherspresentinEPXfortwomainreasons.Thefirst reasonistheuseofCCFVinthefluidsub-domain,andthesecondrea- sonisthepossibilityforthetestspecimentoundergolargerotations andlargedeformation.
Thestructureandthefluidaremeshedindependently,andthen,the structuralmeshissimplyembeddedinto(i.e.,superposedto)thefluid mesh,asshowninFig.2a.Thisdramaticallysimplifiespreparationof thenumericalmodelcomparedwithother(mesh-conforming)FSItech- niques,butitrequiresmoreCPU-intensivecalculationsandmayslightly reducetheaccuracyoftheresultsforagivensizeofthemesh.However, thistechniqueismostfavourableinthecaseoflargedeformationsofthe structure,seeRef.[34].
TheFLSWtechniquefollowsaso-calledweakcouplingbasedondi- rectapplicationoffluidforcestothestructure.Thefluidforcesarise
fromthefluidpressurecomputedattheCCFVcentroids.FLSWisthe most natural choicewhenusing theembeddedapproachandCCFVs forthefluidsub-domain[33].Thisisopposedtotheso-calledstrong couplingofotherFSItechniquesbaseduponconstraints(viaLagrange multipliers)onthevelocitiesatthefluidnodes,whichisthepreferred approachwhenFEsareusedforthefluid.FLSWoperatesonthenumer- icalfluxesofmassandenergyatCCFVinterfacesinteractingwiththe structure.Thesefluxesareblocked(Fig.2b)inordertopreventspuri- ouspassage(leakage)offluidacrossthestructure,aslongastheplate doesnotfail.Thisproducesasortof(weak)feedbackonthefluidflow, duetothepresenceofthestructure.Thefluidforcesareassembledwith otherpotentialexternalforces(seetheAppendixand𝑭extinEq.(2))and subsequentlyusedtocalculatethedynamicequilibriumofthestructure.
WithreferencetoFig.2,inordertodeterminetheportionsoffluid (thinregularmesh)interactingwiththestructure(thicksolidlines),the so-called structuralinfluencedomainis considered(greyzone).Each CCFVinterface(smallhollowsquare)locatedinsidetheinfluencedo- maintransmitsaloadtothenearestpointofthestructureproportional tothepressuredropbetweenthetwofluidcellsformingtheinterface.
Acrucialpartofthealgorithmisthefastupdateofthestructuralinflu- encedomainandthefastsearchfortheinteractingfluidentities(CCFV interfacesinthiscase)ateachtimestepofthenumericalsimulation.
Recentadvancements[35–38,40]inEPXallowforautomaticadaptive meshrefinement(AMR)nearthefluid-structureinterface(FSI-driven adaptationoffluidmesh),whichimprovestheaccuracyoftheembed- dedapproach.AsshowninFig.2c(withjustonelevelofadaptivere- finementforsimplicity),byreducingthesizeofthefluidcellscloseto thestructure,thethicknessofthestructuralinfluencedomaincanbe reducedaccordingly,thusincreasingtheaccuracyoftheembeddedFSI algorithm.
3.2. Numericalmodels
A fully coupled FSI model using one quarter of the experimen- talsetup(byexploitinglongitudinalsymmetries)wasestablished(see Fig. 3c). The steelplate anddiaphragmswere modelledusing aLa- grangiandiscretizationwithReissner-Mindlinshellelements(quadran- glesQ4GSandtrianglesT3GS).Ameshconvergencestudyshowedthat ameshsizeof10mmwasadequatetoreproducetheobservedglobal deformation.Thesteelplateanddiaphragmswerethereforemodelled withanelementsizeofapproximately6mmand10mm,respectively,as thebasemeshpriortoadaptiverefinement(whereAMRwasappliedto thediaphragms).ThediaphragmsweremodelledusingonlyQ4GSele- ments,whilethesteelplatewasrepresentedusingbothQ4GSandT3GS.
Q4GSisa4-nodeelementwith6dofspernodeand20integrationpoints (4intheplane,5throughthethickness),andT3GSisa3-nodeelement with5integrationpoints(1intheplane,5throughthethickness).Sim- plifiedboundaryconditionswereusedforthediaphragms,while the steelplateincludedthecompleteclampingassembly.Thus,onlytheex- posedareaofthediaphragmswasmodelledandallthenodeslocated alongtheperimeterwerefullyfixedagainsttranslationinalldirections (Fig.3a).Theimportanceofincludingthediaphragmfailureprocess inthesimulationofblastwavepropagationinshocktubeswasillus- tratedbyAndreottietal.[47].Itwasfoundthatthediaphragmfailure
Fig.3. Numericalmodel(1/4)ofthe(a)map- pingsimulation, (b)fluidsub-domaininthe firstpartoftheuncoupledapproach(seecase B2 in Fig.4) and(c) fully coupledFSIap- proach.Theplateassemblyshownin(c)was alsoused(stand-alone)inthesecondpartof theuncoupledapproach.
processintroducesa multi-dimensionalflow fielddownstreamof the diaphragms,whichwasobservedasalossofdirectionalenergyinthe distantflowfield.Thediaphragmfailureprocesswillthereforeaffect thereflectedoverpressureonthesteelplateslocatedattherearendof thetube.
Thefluidsub-domainwaspartlydiscretizedby1Dfinitevolumes (segmentsofTUVF)andpartlyby3Dfinitevolumes(bricksofCUVF);
seeFig.3. Aninitialmeshsizeof10 mmwasusedintheentire1D- domain,inthefirstpartofthe3D-domainandinthevicinityofthe plate, according to the mesh sensitivity study in Ref. [39,41]. The cellsizein the tank wasincreased up to80 mm towards theinter- nal walls.This resulted in 1210 TUVFs and190,527base CUVFs in thefluidsub-domainbeforeAMRapplication(whereappropriate).The motivationfor usingTUVFsin-betweenthetwo regions withCUVFs was twofold:toreduce theCPU costby reducing thenumber of fi- nitevolumesandtoenabletheuseofthePAROdirectiveinEPX[44]. ThePAROdirectiveallowsaccountingforfrictionandheatexchange
againsttheinteriorwallsofthetubebyspecifyingtheaverageroughness (0.4micrometres).
The3Dmeshofthefluidsub-domainstartswithacircularcross- sectioninthedriverandfiringsections.Then,atransitionpartof0.6m lengthfollowsimmediatelydownstreamthediaphragms,whichstarts withthecircularcross-sectioninthedriverandendswiththesquare cross-sectioninthedrivensectionthroughoutthefollowing3.3mofthe shocktube(seeFig.3).This3Dpartofthemeshisthenfollowedbya1D partuntilreaching0.6mupstreamofthetestspecimen,withsuitable couplingsbetweenthe1Dand3Dpartsofthefluidmesh(TUBMjunc- tionelements).Thatis,theTUBMconnectsthe1Dpartofthefluidusing TUVFstothefacesoftheneighbouringCUVFsinthe3Dpartofthefluid.
Itisemphasizedthatthelocationofthejunctionelementshouldcoin- cidewithauniformfluidfieldatthispointinthemodel.Thefluidsub- domainconsidersthecomputationalmeshfixed(Eulerianformulation), while thefluid(particles)movesrelativetothesegridpoints.CCFVs wereused,andthenumericalfluxesbetweenadjacentCCFVswerecal-
culatedusingtheapproximateHarten-Lax-van-Leer-Contact(HLLC)Rie- mannsolver[24,44],wherestabilityintheconvectionphaseoftheex- plicitsolutionintimewasensuredbyusingaCourant-Friedrichs-Lewy (CFL)coefficientof0.4.CCFVshaveaninherentrigid-wallboundary conditionduetotheintegralformofthegoverningequations,where thefluxatthecellboundaryisblockedifthereisnoneighbouringcell.
Thus,there isno needtoexplicitlyimposeanyboundaryconditions ontheCCFVsadjacenttotheshocktubewalls.TheHLLCsolverwas chosenduetoitsfavourablecharacteristicswithrespecttolimitingthe numericaldiffusionwhencomputingthefluxesattheinterfacesofthe CCFVs.TheVanLeer-Hancockpredictor-correctorschemewasutilized intheCCFVtoachievesecond-orderaccuracyintime.Secondorderin spacewasreachedviatheGreen-Gaussreconstructionoftheconserva- tivevariablesusingaDuboislimitertomitigatethepossibilityofnumer- icalinstabilitiesattheshockfrontwherethesolutionisdiscontinuous.
Theresultingnumericalschemeistotalvariationdiminishing(TVD).A sensitivitystudywascarriedouttoevaluatetheinfluenceoftheDubois limitercoefficient[44],anumberbetween0and1,withhighervalues correspondingtomoreaccuratebutalsopotentiallyunstableresults.It wasfoundthatthedefaultvalueusedinEPX(0.50)gaveequallygood resultsasothermoreaggressivevalues(0.75).
Basedontheinformationfrom themanufacturer,theMelinex di- aphragmswereassumedtobehaveelasto-plasticallyuntilfracture.De- pendingonthethicknessofthediaphragms,theyieldstressandplastic moduluswereintherangeof100–160 MPaand13.9–54.7MPa,re- spectively.Fracturewasmodelledusingelementerosionandwasiniti- atedwhenalltheintegrationpointsintherespectiveelementreached acriticalvalueof100%forthemaximumprinciplestrain.Inanat- tempttopredictthecrackpropagationobservedintheexperiments,use wasmadeofAMRofthediaphragmsdrivenbytheaccumulatedplas- ticstrain𝑝.Recentadvancements[35]inEPXallowforAMRbasedon user-definedcriteria,whichmakesitconvenienttorelatetheAMRto theplasticstrain.Thatis,themeshrefinementoccursatuser-defined levelsoftheplasticstrainandatsuccessivelevelsofrefinement.This studyuseduptotwosuccessiverefinementswithintherangeof0.01
≤𝑝≤0.4,resultinginaminimumelementsizeof2.5mmwhenp>
0.4.Ductilefractureofthediaphragmscouldthenbepredictedwithout toomuchlossofmasswhenusingelementerosionincombinationwith AMR.
Thediaphragmsarecompletelydecoupledfromthefluidduringthe fillingprocess.Thatis,thediaphragmsarefirstloadedbyanexternally imposedpressuresimilartothatofthecompressedairinthedriver(see Table1).Onceequilibriumisreachedaroundthedeformedconfigura- tion,theexternallyimposedpressureisremovedandthefluidstates areinitializedinthedriver,firinganddrivenchambers.Thepressure gradientsoverthediaphragmswillthenensureequilibriumuntilrapid ventingofthefiringsectioninitiatesthediaphragmfractureprocess.An embeddedFSItechnique(FLSW)[34,44]andFSI-drivenAMRwasused forthecouplingatthefluid-diaphragm(F-D)interface.FSI-drivenAMR wasactivatedinthefluidsub-domaintoobtainasufficientlyrefined fluidmeshattheF-Dinterface.
TheFLSWFSItechniqueandFSI-drivenAMRwerealsousedinthe fluidforthecouplingatthefluid-structure(F-S)interfacebetweenthe fluidandthesteelplate.EPXthenenablesautomaticrefinementofthe fluidmeshinthevicinityoftheplatethatcanmoveandundergolarge deformations.Thisallowsforasufficientlyfinefluidmeshsizetorep- resentthenearinstantaneousriseinpressureovertheblastwave.A meshsensitivitystudyshowedthatameshsizeof5mmwassufficient tocapturethegoverningphysicsinthefluidsub-domain.Thenumberof AMR-generatedelementswasapproximately300,000CUVFinthefluid attheF-Dinterface,10,000Q4GSinthediaphragmsand200,000CUVF inthefluidattheF-Sinterface.Contactisactivatedbetweenthevar- iousdiaphragmsandbetweenthediaphragmsandtubewallstoavoid inter-penetrationduringthediaphragmfailureprocess.
Thesymmetryofthemodelwasalsoexploitedwhenmodellingthe plateandclampingassemblyusinga1/4modelwithsuitablegeometry
constraints(seeFig.3c).Theboltsandtheclampingframeswererep- resentedbysolidelementswith3dofspernodeusingboththe8-node brickelementCUB8with8integrationpointsandthe6-nodewedgeel- ementPR6with6integrationpoints.Materialparametersforthesteel clampingframesweretakenfromRef.[36]usingtheVPJCmodel(see theAppendix,Eq.(7)andTableA.1).Eachoftheboltswaspre-stressed toaninitialtorque(𝑀𝑡=200Nm),resultinginaclampingpressurebe- tweentheframesandtheplate.Thiswasaccountedforbymodelling thelowerclampingframeandboltsasoneinitiallystress-freecompo- nent,whileanexternalpressurewasappliedatthecontactareabetween theboltheadandtheupperclampingframe.Thecontactpressurewas determined usingtheapproachsuggestedin Ref.[36], resultingina contactpressureof44MPaappliedoverthecontactareaforeachnut (1060mm2)throughoutthesimulation.Theclampingpressurewasim- posedvia4-nodeboundaryconditionelementsCL3D.Theseelements automaticallyrecognizethesolidelementstowhichtheyareattached andusetheassignedpressurehistories.Thelowerclampingframewas fullyblockedatitsbacksurface.
Contactbetweentheplate,boltsandframeswasmodelledusinga node-to-surfacecontactalgorithm(GLIS)utilizingslavenodesandmas- tersurfaceswherecontactwasenforcedbyLagrangianmultiplierswhen aslavenodepenetratedamastersurface.Theplatewasmodelledasthe slave,andboththestaticandthedynamicfrictioncoefficientbetween theplateandclampingframesweresetto0.50,atypicalvaluefrom theliteratureforasteel-to-steelinterface.Adetailedpresentationofthe numericalmodellingofthesteelplateandclampingassemblyisfound inAuneetal.[36].Itshouldbeemphasizedthatthemodellingofthe clampingassemblywasessentialtoobtainaccurateplatedeformation.
Thisisbecausetheplateslidessomewhatbetweentheclampingframes whiledeforming.Thus,theplatecannotbesimplyconsideredasfixed alongtheperimeteroftheexposedarea,sincethiswouldresultinan overlyconstrainedbehaviouroftheplate.
3.3. ComputationalmethodologytostudyFSIeffects
Fluid-structureinteraction(FSI)effectsarestudiedbycomparingthe numericalpredictionsoftheuncoupledandcoupledFSIapproaches.A schematicrepresentationofthecomputationalframeworkadoptedin thepresentworkispresentedinFig.4.Thatis,priortothesimulations usingeithertheuncoupledorthecoupledapproach,apreliminarysim- ulationwasperformedusingthenumericalmodelpresentedinFig.3a, includingthedetailedrepresentationofthediaphragmfailureprocess anditsinfluenceontheblastwaveformationalongtheshocktubein eachofthetests(seetextboxAinFig.4).Thissimulationgeneratesa so-calledmapfilecontaining,foreachfluidfinitevolume,thephysical conditionsjustbeforetheshockwavereachestherightendofthe1D fluidsub-domain.Themappingprocedurewaspossiblesinceallthenu- mericalmodelspresentedinFig.3usedtheexactsamefluiddiscretiza- tionthroughout thefirst3Dand1Dpartsof themesh.Fig.5shows thediaphragmfailureprocessinthepreliminarysimulationoftestD35.
BothFSI-driven AMR(inthefluid)andplasticity-basedAMR (inthe diaphragms) areusedforadetailedrepresentationofthediaphragm ruptureanditsinfluenceontheflowfield.
Then,anuncoupledsimulationapproachisperformedconsistingof twosteps(seetextboxBin Fig.4).Thefirststep(B1)isanEulerian (fluid-only)simulationusing themapfile asinitialconditionsin the fluidsub-domainandproducingthepressuretimehistory𝑝(𝑡)onthe rigidwall(platelocation)inFig.3b.Thus,theuncoupledsimulations maketheinherentassumptionthatthepressureisunalteredbythestruc- tural motion,andviceversa.Theuncoupled approachis thereforea conservativesimplificationoftherealbehaviourbecauseitisexpected thatthisapproachwilloverestimatetheactualpressuresincetheplate is notallowedtodeformin theEuleriansimulations.Theuncoupled approachwillthereforebeusedinthefollowingasareferencetoiden- tifyFSIeffectsduringthedynamicresponseofthedeformableplates.
ThereflectedoverpressuresobtainedontherigidwallintheEulerian
Fig.4. Conceptualschemeofthestrategyusedto studyFSIeffectsbyusingthenumericalmodelsin Fig.3.NotethatboththeEuleriansimulationsin theuncoupledapproachandtheFSIsimulationsin thecoupledapproachstartedfromthesameinitial conditionsinthefluidsub-domain(mapfile),en- suringthattheincomingblastwavewasidentical inbothapproaches.
Fig. 5.Illustrationof the preliminarysimu- lationincludingthe diaphragmsfailure pro- cess.Upperviewprovidesthecompleteview ofthepressurefieldfortheentiremodel,while thelowerviewcontainsacloseuponthedi- aphragmsandthefluidinthevicinityofthe diaphragmsimmediatelybeforerupture(left), justafterrupture(middle)andatcompletefail- ure(right). Notethe automaticAMRacting bothonthefluid(FSI-driven)andonthedi- aphragms(plastic-straindriven).
simulationsareshowninFig.6.Thesecondstep(B2)oftheuncoupled approachisaLagrangian(structure-only)simulationincludingonlythe thinsteelplateandtheclampingassemblyshownintherightpartof Fig.3c.Thissimulationusesthepressurehistory𝑝(𝑡)predictedinthe Euleriansimulationasanimposedloadingtoobtainthecorresponding dynamicresponseoftheplate.Analogoustotheclampingpressure,the blastpressureontheexposedareaoftheplatewasmodelledbyCL3D
boundaryconditionelements.Theblastpressurewasimposedasauni- formlydistributedpressureontheexposedareaofthesteelplateinthe uncoupledapproach.TheuniformdistributionwasjustifiedbytheEu- leriansimulation,whichpredictedauniformblastpressureattherigid wall.
Finally,asimulationfollowingthecoupledapproachiscarriedout for eachof thetests in Table1,usingthecorresponding mapfileto
Table2
NumericalresultsintermsofblastpropertiesfromtheEuleriansimulationsandpressuremeasurements𝑝r,maxand thecorrespondingsaturatedimpulse𝑖r+,satatSensor1intheuncoupledandcoupledFSIapproach.TheFSIeffects Δ𝑝r,maxandΔ𝑖r,+arealsogiven.
Test
Blast properties ∗ Pressure measurements at Sensor 1
Eulerian simulations Uncoupled approach Coupled approach FSI effects ∗∗ 𝑀 s 𝑝 r,max 𝑡 d+ 𝑖 r+ 𝑝 r,max,u 𝑖 r+,sat,u 𝑝 r,max,c 𝑖 r+,sat,c Δ𝑝 r,max Δ𝑖 r+,sat
[ - ] [kPa] [ms] [kPa ms] [kPa] [kPa ms] [kPa] [kPa ms] [%] [%]
D05 1.28 258.60 28.5 2659.0 242.0 280.8 231.5 257.9 -4.3 -8.1 D15 1.64 595.90 51.3 8633.3 552.0 503.2 500.3 448.1 -9.4 -11.0 D25 1.74 800.60 74.1 16,361.4 743.0 641.4 663.8 569.2 -10.7 -11.3 D35 1.84 1097.50 86.4 23,513.6 1045.2 876.7 916.5 747.5 -12.3 -14.7 D60 2.09 1462.00 89.0 36,951.5 1401.7 1091.9 1191.8 951.6 -15.0 -12.8
∗Blast parameters representing the pressure histories at the rigid wall in Fig. 3b. ∗∗Δ𝑝r,max=(𝑝r,max,c− 𝑝r,max,u)∕(𝑝r,max,u)× 100%,Δ𝑖r+,sat=(𝑖r+,sat,c−𝑖r+,sat,u)∕(𝑖r+,sat,u)× 100%.
Fig.6.PressurecurvesobtainedontherigidwallintheEuleriansimulations (seeFig.3b).Thesepressurehistoriesareusedtoloadtheplatesinthepurely Lagrangiansimulations(uncoupledapproach).Notethatthecurvesareshifted intimeforimprovedreadability.Eachcurvewasright-shifted3mswithrespect tothepreviousone.
settheinitialconditionsinthefluidsub-domainandincludingboththe fluidandstructuralsub-domaininthesamesimulation(seetextboxCin Fig.4).ItisimportanttoemphasizethatboththeEuleriansimulation intheuncoupledapproachandtheFSIsimulationinthecoupledap- proachstartedfromthesameinitialconditionsinthefluidsub-domain (mapfile),ensuringthattheincomingblastwaveisidenticalinboth approaches.ThisallowsfornumericalinvestigationsonFSIeffectsdur- ingthedynamic responseoftheplatesbasedontheoutputlistedin textboxD.Theresultsfromthefullycoupledsimulationsaretherefore comparedtothecorrespondingresultsobtainedwiththeuncoupledFSI approach.Experimentalresultswillfinallybecomparedtothefullycou- pledsimulationsinSection4toensurethatthenumericalpredictions arereasonable.
3.4. QuantificationofFSIeffects
Inblast-resistantdesign,thereflectedoverpressureistypicallyrep- resentedasapressurehistory𝑝(𝑡)describedbythepeakreflectedover- pressure𝑝r,max,thedurationofthepositivephase𝑡d+andthepositive specificimpulse𝑖r+.TheMachnumber𝑀sisalsofrequentlyusedtoin- dicatetheblastintensity.Theseblastparametersarethereforelistedin Table2forcompletenessintheevaluationoftheblastpropertiesinthe shocktubetests.Theblastparameterswereobtainedfromthepressure historiesontherigidwallinthepurelyEuleriansimulations(seeFig.6).
Table3
NumericalresultsfromtheuncoupledandcoupledFSIapproachintermsof mid-pointdeflections,saturateddurations𝑡d+,satandtheFSIeffectΔ𝑑z,max.
Test
Mid-point deflections and saturated time ∗
Uncoupled approach Coupled approach FSI effect ∗∗ 𝑑 z,max,u 𝑑 z,p,u 𝑡 d+,sat,u 𝑑 z,max,c 𝑑 z,p,c 𝑡 d+,sat,c Δ𝑑 z,max
[mm] [mm] [ms] [mm] [mm] [ms] [%]
D05 16.5 13.6 1.26 15.8 12.9 1.21 -4.1 D15 26.9 25.3 0.97 24.9 23.1 0.94 -7.5 D25 32.9 31.3 0.92 29.7 28.2 0.89 -9.5 D35 41.0 39.7 0.88 36.3 34.9 0.84 -11.4 D60 50.4 49.2 0.82 43.4 42.2 0.79 -13.8
∗𝑡d+,sat=thetimeittakesfromthestartofplatemovementuntilpermanent deformation𝑑z,p.∗∗Δ𝑑z,max=(𝑑z,max,c−𝑑z,max,u)∕(𝑑z,max,u)× 100%.
Itisimportanttoemphasizethattheseshocktubetestswerefound tobeinthedynamicloadingdomainbyAuneetal.[36].Thisclassifi- cationwasbasedontheratiobetweenthepositivephaseduration𝑡d+
andthenaturalperiodofvibration𝑇𝑛intheplates.Thenaturalperiod ofvibration𝑇𝑛 wasestimatedtobe12.5ms,resultinginratios𝑡d+∕𝑇𝑛 rangingfrom2.3to7.1forthetestslistedinTable2.Hence,itisnot thetotalimpulse𝑖r+oftheloadingthatgovernsthedynamicresponse butrathertheprofileoftheloadinghistoryduetotheoverlappingof theplateresponsewiththepositivephaseduration𝑡d+(see,e.g.,[48]).
Therefore,inthedynamicloadingdomain,itisonlythesaturatedpart oftheimpulse𝑖r+,satthatcontributestotheplatedeformation.Focusing onthetotalreflectedimpulse 𝑖r+ mightbe misleadingsinceonlythe saturatedimpulse𝑖r+,sat isresponsibleforthepermanentdeformation oftheplates.Thisworkadoptsthedefinitionofthesaturatedimpulse 𝑖r+,satassuggestedintheworkbyBaietal.[4],i.e.,thereflectedimpulse untilthesaturatedduration𝑡d+,satthatcorrespondstothetimeofperma- nentdeformation𝑑z,pintheplates.Thesaturatedduration𝑡d+,satthere- forecorrespondstotheperiodoftimeinwhichtheplateexperiences permanentdeformations.Beyondthisperiod,themotionof theplate ceasesandtheplatemainlyundergoesminorelasticvibrationsaround itspermanentdeformedconfiguration.Thesaturationphenomenonwill befurtheraddressedwhendiscussingtheresultspresentedinFig.7.
Fig.7a-bcomparethemid-pointdeflectionsandmid-pointveloci- ties (obtainedbydifferentiatingthedeflectionsintime) intheplate, respectively, forthecoupledapproach tothosein theuncoupled ap- proach,whileFig.7ccontainsthepressurehistoriesatthecomputa- tionalcell(CUVF)closesttothepointwhereSensor1waslocatedin theexperiments.Maximummid-pointdeflections𝑑z,max,permanentmid- pointdeflections𝑑z,pandthesaturatedduration𝑡d+,sataresummarized andcomparedtotheuncoupledapproachinTable3.Negativevaluesof thedifferenceinmaximumdeflectionΔ𝑑z,maximplythatthemaximum mid-pointdeflectionsarelargerintheuncoupledapproach.Itshould
Fig.7. InvestigationofFSIeffectsonthe(a) mid-point deflections, (b) the induced mid- pointvelocityintheplates,(c)pressuresmea- suredatSensor1and(d)thesamedataas(c) wherethetimeaxisismorefocusedonthere- flectedoverpressure.Notethatthecurvesin(c) and(d)areshiftedintimeforimprovedread- ability.Eachcurvewasright-shifted3mswith respecttothepreviousone.
benotedthatFig.7dcontainsthesamedataasFig.7c;however,the timeaxisismorefocusedonthereflectedoverpressureatSensor1.The mainpurposeofFig.7dis thereforetoillustratetheinfluenceofFSI onthepeakreflectedoverpressure.Inaddition,notethatthesimula- tionsfollowingthecoupledapproachwereintentionallystoppedearlier thanthecorrespondinguncoupledsimulations(seeFig.7c).Thiswas because,ontheonehand,thedynamicresponseofinterestwasalready reachedatthispointintime,while,ontheotherhand,theremaining partofthecoupledsimulationwouldrequireasignificantCPUcostdue tothedecreaseinthecriticaltimestepcausedbytheFSI-drivenAMR inthefluidclosetotheplate.
Asexpected,theuncoupledapproachpredictslargerdeformations than the corresponding fully coupled FSI simulation (Fig. 7a and Table3).Itisobservedthatthemid-pointdeflectionsarereducedby approximately 4-14%when consideringFSI.The cleartrendis that higherpressuremagnitudesresultin increasedFSIeffectsduringthe dynamicresponseoftheplates.Thesametrendwasobservedforthein- ducedvelocitiesintheplates(Fig.7b).Fig.7c-dshowareductioninthe initialpeakreflectedoverpressureinthecoupledsimulations,wherea slighttrendofanincreasedreductionatincreasingpressuremagnitudes isobserved(Fig.7d).Thatis,thepressuremeasuredatSensor1inthe uncoupledsimulationsisalwayshigherthanthatinthecoupledFSIap- proach.ItisimportanttoemphasizethatSensor1islocated24.5cm upstreamofthetestspecimen(seeFig.1a)andthatthepeakpressure immediatelyafterreflectionisassumedtobeindependentofthestiff- nessofthestructure(see,e.g.,[25]).TheobservedFSIeffectsatSensor 1aresummarizedinTable2,providingthepeakreflectedoverpressure 𝑝r,maxandthesaturatedimpulse𝑖r+,satforthecoupledandtheuncoupled approach.Thesaturatedimpulse𝑖r+,satisfoundbyusingthepressurehis- toryatSensor1.Theintegralofthereflectedoverpressureistakenfrom thetimeofarrivalofthereflectedpressureatSensor1(seesecondjump inpressureinFig.7d)overthetimeinterval𝑡d+,sat,whichcorresponds tothetimeintervalofthedynamicresponsebeforetheplatereachesits permanentdeformation.Thesametrendisobservedfortheloadingas
forthemid-pointdeflections,i.e.,boththepeakreflectedoverpressure andthesaturatedimpulsearereducedinthecoupledsimulations.More- over,higherblastpressuremagnitudesresultinincreasedFSIeffects.
Fig.7c-dalsoshowthattheincident(side-on)pressureswereinex- cellent agreement,indicatingthatthereduced reflectedoverpressure maybeduetothedeformationoftheplates.Thiswasalsoobserved inpreviousstudies,whichindicatedthattheblastmitigationcouldbe relatedtotheinducedvelocityintheplate(see,e.g.,[23–25]),while Hanssenetal.[26]suggestedthatthereductioninreflectedpressureis duetothedeformedshapeoftheplatewhichresemblesaglobaldome.
Hanssenetal.[26]argued thatthedeformed shapeproducesanon- uniformspatialandtemporaldistributionofthepressureinthevicinity oftheplate.Thereductioninreflectedpressureseemstooccurovera periodintimethatissimilartothesaturateddurations𝑡d+,satlistedin Table3.Then,verylimitedFSIeffectsareobservedthroughoutthere- mainingpartofthepositive phase(Fig.7c). Thismakesitnaturalto relatethereductioninreflectedpressuretotheinducedvelocityinthe plate.
Fig.8showsacomparisonoftheplatedeformationprofilesin an attempttoinvestigatetheinfluenceofFSIeffectsonthedeformedshape oftheplates.Thedeformationprofilesareextractedatmagnitudesof0
%,25%,50%,75%and100%ofmaximummid-pointdeflectionin boththeuncoupledandcoupledsimulation.Thecomparisonislimited totestD35,sincethesametrendwasobservedinalltests.
Asexpected,thedeformationprofilesshow asimilartrendtothe mid-point deflectionsasin Fig.7a. Thatis,the uncoupledapproach predictslargerdeformationsthanthecorrespondingfullycoupledFSI simulations.Thetrendisthathigherdeformationmagnitudesresultin increasedFSIeffectsduringthedynamicresponseoftheplates.Itisalso notedthatthedeformationprofilesinthetwoapproacheshavemoreor lessthesameshapeatthesamelevelofdeflection,butwithdifferent magnitudes.Bothapproachesshowthecharacteristicbehaviourofblast- loadedplates,whereplasticyieldlinesarefirstformednearthesupport andthentraveltowardsthecentreoftheplate(see,e.g.,[49]).Thisin-
Fig.8. Comparisonofdeformationprofilesat0%,25%,50%,75%and100
%ofmaximumdeflectionfortheuncoupledandcoupledapproachintestD35.
Thedeformationprofilesareextractedfromthecentrealongthex-axis.
dicatesthatthemid-pointvelocitiesinFig.7bmaybeagoodestimate oftheinducedvelocityintheplateduringtheFSI.Thus,themid-point velocitiesarerepresentativeforthestraight,horizontalpartoftheplate locatedin-betweentheplasticyieldlinesuntiltheymeetatthecentre.
Thisbuildsconfidenceinthefactthattheinducedvelocityintheplate producestheobservedpressuredropinfrontoftheplate(Fig.7c-d), whichisalsoinaccordancewithstudiesonFSIeffectsduringthere- sponseoffree-standingplates(see,e.g.,[23–25]).
Fig.9containsacomparison ofthepredictedoverpressure inthe vicinityoftheplateintheuncoupledandthecoupledFSIapproach.
ThecomparisoniscarriedoutatcharacteristictimesintestD35and showsthepressureactingontheloadedsurfaceoftheplateandthe fullviewofalongitudinalcross-sectioninthecenterofthetube.This viewof thecross-sectionis obtainedbymirroring thequarter-model acrossonesymmetryplane,andthisviewisonlyusedforvisualization purposesofthepressuredistributionneartheF-Sinterface.Notethatthe tankwasnotpresentintheuncoupledapproach(seeFig.3b);however, the‘virtual’contourofthetankisincludedintheuncoupledcontour mapsasagreyshadedlinetoindicatethecorrespondingfluidvolume surroundingtheplateinthecoupledsimulation.
Asexpectedfromarigidreflectingsurface,theuncoupledsimulation showsplanar,uniformwavefrontsthroughouttheentiresimulation(left partofFig.9a–f).Aplanar,uniformpressurewaveisalsoobservedin thecoupledsimulationimmediatelyaftertheinitialreflectionwhenthe reflectedshockwavestarts travellingfromrighttoleft(rightpartof Fig.9a).Then,astheplatestarts movinginthecoupledsimulations, thedeformationofthesteelplate inducesanon-uniformspatialand temporaldistributionof thepressureneartheplate(Fig.9b–e). This isfirstobservedasareducedpressureinthecentralpartoftheplate (Fig.9b),resultinginpressurewavesthatpropagatebothradiallyand totheleft(Fig.9c–e).Astheplatedeforms,itundertakesacurvedshape thatseemstoresultinafocusingeffectofthepressureinthecentral partoftheplate(Fig.9d–e).Thereisatrendofareductioninreflected pressurebeforethemaximumdeformationisreachedatapproximately 𝑡=1.14ms(Fig.9b–c).Then,anincreaseinmaximumpressureisob- servedinFig.9dat𝑡=1.20ms.Theradialpressurewavescontinueuntil theendoftheelasticrebound(Fig.9e)whentheplatereachesitsper- manentdeformation.Thereafter,onlylimitedFSIeffectsareobserved throughouttheremainingpartofthesimulation(seeFig.9f).
Thismakesitnaturaltorelatethereductionofmid-pointdeflection inFig.7atoboththedeformedshapeandtheinducedvelocityinthe plate,which altersthereflectedpressureinthevicinityof theplate.
Thatis,theinitialreductioninreflectedpressureisrelatedtothein- ducedvelocityoftheplate(rightpartinFig.9b),whilethesubsequent
increaseinpeakreflectedpressuremaybeduetothedeformedshape oftheplatethatinducesanon-uniformpressuredistribution(pressure focusingeffect)inthecentreoftheplate(seerightpartinFig.9d).Itis interestingtonotethattheincreaseinpressuremagnitudeatmaximum deflectioninthefullycoupledapproach(rightpartofFig.9d)islarger thanthatintheuncoupledapproach(leftpartofFig.9d).
ThetrendshownfortestD35inFig.9isrepresentativeofalltests underconsiderationinthisstudy.Thatis,theeffectoftheinducedveloc- ityintheplatetendstoreducethepressure,whilethedeformedshape oftheplateinducespressuremagnitudeslargerthanthoseintheEule- riansimulations(pressurefocusingeffect).Itisinterestingtonotethat theincreaseinpressuremagnitudesoccursmoreorlessattheinstant ofmaximumdeformationintheplate.Totheauthors’bestknowledge, therearenopreviousstudiesonclampedsteelplatesthatobservethis typeofFSIeffectintermsofincreasedpressureduetothedeformed shapeoftheplate(Fig.9d).
3.5. InterpretationoftheFSIeffects
ThisinterpretationofthepressurewavesoccurringduringtheFSI partofthedynamicresponseiscorroboratedinFig.10byamorede- tailedinvestigationof theresulting wavepatterns.Fig.10shows the densitygradienttovisualizethevariationsintheresultingwavepattern closetotheplate.ThistypeofvisualizationissimilartotheSchlieren techniquethatisoftenusedinexperimentstorepresentsmalldifferences inpressure,i.e.,thelocationandmagnitudeofexpansionandcompres- sionregionsinafluidflow(see,e.g.,[50]).Theuncoupledandthecou- pledFSIapproachareshownintheleftandrightcolumn,respectively, ofthefigureforeachinstantofinterest.Notethatthechosentimesof interestaresimilartothoseinFig.9exceptthatt=1.90msisreplaced byt=0.80mstoobtainmoreinsightintothewavepatternsduringthe initialphaseoftheplateresponse.Toobtainareasonableresolutionof thedensitygradient,itwasnecessarytouseanevenfinermeshthan thatintheremainingpartsofthisstudy.Thefluidmeshwastherefore refinedevenfurtherbyusingtwoadditionallevelsofFSI-drivenAMR inthefluidsub-domain.Thisresultedinameshsizeof1.25mminthe fluidsurroundingtheplate(seeFig.10a).Notethattheadditionalre- finementwasusefultoimproveflowvisualization,butithadnovisible effectsontheresultspresentedsofarconcerningtheplatedeformation andthepressuretimehistories.
Thereflectedshockwavecanbeseenastheplanarwavewithadis- tinctdensitygradientpropagatingfromrighttoleft.Asexpected,itis observedthattheshockwavereflectsonaplanarsurfaceinboththeun- coupledandcoupledsimulation(Fig.10a).Thus,theplanarnatureof theshockwaveisnotalteredduringorafterthereflectionitself.How- ever,assoonastheplatestartstoundergodeformations,itisevident thattheFSIintroducesasignificantlymorecomplexwavepatterninthe regionbehindthereflectedshockwave(seeFig.10b–f).Thisispartic- ularlyevidentwhencomparingtheseobservationstothoseintheun- coupledsimulation,wheretheshockwavereflectsonanon-deformable surface.TheFSIgeneratesaseriesofcompressionandexpansionwaves propagatingbothin thelongitudinalandinthetransversaldirection of thetube.Eventually,thetransversalwavesmeetinthecentre,re- sultinginafocusingofthecompressionwavesattheplatecentre(see Fig.10e),whichthenresultsintheincreaseandfocusingofthepres- sureasobservedinFig.9d.Itisinterestingtonotethatthispointin timecorrespondstotheelasticreboundoftheplate,immediatelyafter thetimeofpeakdeflection(𝑡=1.14ms).
Then,astheplatereachesitspermanentdeformedconfigurationat 𝑡=1.25ms,itisobservedthatsomesmalldisturbancesaregenerated behindtheplateastheplateundergoeselasticoscillationsaroundthe permanentdeflection(seeFig.10e–f).Thesesmalldisturbancesseemto initiateduringtheelasticreboundoftheplate(seeFig.10e)andthen propagatealongtheplatesurfaceasaresultofconservationofmomen- tumintheair(seeFig.10f).However,despitetherelativelystrongden- sitygradientofthesedisturbances,thislastphaseoftheFSIhaslimited
Fig.9.Comparisonoftheuncoupled(left)and coupled(right)FSIapproachatcharacteristic timesintestD35.Fringecoloursrepresentthe contourmapoftheoverpressure(inkPa)inthe vicinityandontheplate.Cross-sectionalviews alongthecentreofthefluidareshowntoen- ableaclearviewofthefluid-structureinter- face.Timezero(𝑡=0)istakenasthearrivalof theshockwaveatSensor1locatedupstream ofthetestspecimen(seeFig.1a).
influenceonthepressurefielddownstreamoftheplate(seerightside ofplateinFig.9d–e).
Itshouldalsobenotedthatastronggradientisobserved(darkblue zone)overthethicknessoftheplateinallofthecoupledsimulations.
Thiszoneofhighgradientcoincideswiththeinfluencedomainofthe FLSWalgorithm(seeFig.2)andrepresentsthesuddenchangeinpres- sureanddensityacrosstheplate.
ThedetailedinvestigationofthedensitygradientinFig.10confirms thatthepressurewavesinfrontoftheplateareduetotheinducedmo- tionanddeformedshapeoftheplate.Aseriesofplanarcompressionand
expansionwavesinitiatefromthecentralpartoftheplate(seeFig.10b–
d),whereplasticyieldlinesformaplanarareathatreducesinsizeas theyieldlinespropagatetowardstheplatecentre(Fig.8).Thesepres- surewavesresultinapressuredropinfrontoftheplate(Fig.9a–c).
Atthesametime,radialwavesinitiateatthelocationoftheyieldlines surroundingthisplanararea.Eventually,theseradialwavesmeetatthe platecentre(seeFig.10e),producingafocusingeffectthatcorresponds tothepressureincreaseinFig.9d.
Fig. 11 illustrates additional results on the exposed plate area (Fig.11a)andtheapproachusedtoobtainthepressuresactingonthe
Fig.10. Visualizationofpressurewavesgen- eratedduringtheuncoupled(left)andcoupled (right)FSIapproachatcharacteristictimesin testD35.Fringecoloursrepresentthecontour mapofthedensity gradientmagnitude.The fullyrefinedfluidmeshisalsoshownonone quarterofthemodelin(a).Sensor1islocated attheroofofthetubeandatthepositionof theleftverticaledgeofeachimage.
plate(Fig.11b) inconjunction withtheembedded(FLSW) FSItech- nique. The pressure acting on the plate is obtained using the EFSI functionalityinEPX.This basicallyextracts thefluidpressure inthe closestmeaningfulfluidelement,i.e.,bydisregardingelementswithin thestructural influencedomain (see Fig.2).Fig. 11b givesa close- up of theF-Sinterface in therightpartof Fig.9d,where theplate isillustratedas athick blackline. Thewidthofthestructuralinflu- encedomain isindicated by𝐷 andis in theorder of twofluidele- ments(atthemaximumrefinementleveloftheFSI-drivenAMR).The fluidpressureontheblast-exposedareaoftheplateisextractedfrom the surfacelabelled as ’EFSI plate’ andrepresented by a thick dot- tedlineinFig.11b.This EFSIsurfaceislocated atacertaindistance
𝑑 fromtheactualpositionof theplatebecausethenumericalfluxes areblockedatallCCFV interfaceswithinthestructuralinfluencedo- main(seealsoFig.2).Thisimpliesthatthepressureandotherphys- ical quantitiesin the fluidelementslocated inside theinfluencedo- main,arenotmeaningfulphysicallyandthusnotrepresentativeofthe actual pressure acting on the plate.Note the smeared pressure gra- dient across the plate in Fig.11b,observed asa gradual change in fringevaluesfromyellowtoblueacrosstheplate.Thisalsoillustrates the interestof using FSI-driven AMR in order to reduce the sizeof the influencedomain andof increasingthe accuracy of theembed- dedFSIalgorithmtoobtainmoreprecisepressuredistributionsonthe plate.
Fig.11. Illustrationofthepressureloadingin thevicinityoftheplateintestD35at𝑡=1.00 ms: (a)cross-sectionalviewof surfaceover- pressureactingontheexposedareaoftheplate (sameasinFig.9c),and(b)close-uponthe influencedomaininthecross-sectionalview.
Fringe coloursrepresentthecontourmapof theoverpressure(inkPa)inthevicinityofthe plate,wherethecolourbaristhesameasin Fig.9.
4. Experimentalvalidation
AsalreadystatedinSection3.3,thestudiesonFSIeffectswerecon- ductedbypurelynumericalinvestigationsinthiswork.Thatis,nore- sultsfromtheexperimentalcampaignwereusedinpreparingorcali- bratingthenumericalsimulations.Thechosencomputationalstrategy ensuredthattheincomingblastwavewasperfectlyidenticalinboththe uncoupledandcoupledapproach,toenablebothqualitativeandquan- titativestudies,withinthelimitationsofthenumericalmodel,onthe influenceofFSIeffectsonthedynamicresponseofblast-loadedsteel plates.However,fromanengineeringpointofview,comparisonwith experimentaldataisalwaysofinteresttoevaluatethepredictivecapa- bilitiesofthenumericalmethodologyincapturingtheactualphysicsof theproblem.Theresultsfromthefullycoupledsimulationsarethere- forecomparedtothecorrespondingresultsobtainedfromtheexperi- mentspresentedinSection2.Theexperimentalvalidationwasfocused onthreeperformanceindicatorsintermsofthestructuralresponse,i.e., themid-pointdeflections,themid-pointvelocitiesandthedeformation profiles.Inaddition,thepressuremeasurementsatSensor1wereused toassesstheperformanceofthefluidsub-domaininpredictingtheblast loading.
Fig.12a-bcomparethemid-pointdeflectionsandmid-pointveloci- tiesintheplate,respectively,forthecoupledapproachtothoseinthe experiments,whileFig.12c-dcontainthecorrespondingpressurehis- toriesatSensor1.ItshouldbenotedthatFig.12dcontainsthesame dataasFig.12c,butwithacloserviewontheinitialpressuremeasure- mentsatSensor1.ThemainpurposeofFig.12disthereforetocompare thenumericalpredictionsofthefullycoupledapproachwiththecor- respondingmeasurementsintheexperimentsintermsofincomingand reflectedoverpressures.Fig.13 comparesthedeformationprofilesat magnitudesof0%,25%,50%,75%and100%ofmaximummid- pointdeflectioninboththecoupledsimulationandtheexperiment.As inFig.8,thecomparisonislimitedtotestD35sincethesametrendwas observedinalltests.Itshouldbenotedthatsuchcomparisonsofdefor- mationprofilesshouldbecarriedoutwithcaution.Duetothediscrete nature(i.e.,temporaldiscretization)ofthedatasamplinginboththe numericalsolutionsandintheexperiments,itcanbedifficulttocom- paretheprofilesattheexactsamemagnitudeofdeformation.Thefast dynamicnaturefrom0%to100%ofmaximummid-pointdeflection, whichtypicallytakesplaceinlessthan1ms,makestheextractedde- formationprofilesquitesensitivetotheexacttimeinstantchosenfor visualization.Despitethisinherentsensitivityofthesecomparisons,the
comparisonofdeformationprofilesisvaluableinevaluatingthepredic- tivecapabilitiesofthenumericalmodel.Toreducetheinherenterrorin thecomparisonbecausethesamplinginstantsintheexperimentsdonot exactlycoincidewiththoseinthesimulations,thesimulatedprofilesare linearlyinterpolatedbetweenthetwonearestsamplinginstants.
Thefullycoupledsimulationsweregenerallyingoodagreementwith theexperimentaldata.Excellentagreementisobtainedwithrespectto mid-pointdeflections(Fig.12a)andmid-pointvelocities(Fig.12b)for testsD05toD35,whilethesimulationoftestD60seemstoslightlyun- derestimatethedeflectionsandvelocities.Thedeformationprofilesare alsoinverygoodagreementwiththeexperimentalobservations(see Fig.13).Alargerdiscrepancybetweencomputedandexperimentalpro- filesisobservedforthelowestvaluesofdeflection(0%and25%),which isasexpectedgiventhedifficultyofmeasuringsuchsmallvaluesunder highlydynamicconditions.Acomparisonof thepressurehistoriesat Sensor1showsanexcellentagreementinthetimeofarrivalofthere- flectedshockwave,indicatingthatthevelocityofthatwaveiswellcap- turedbythenumericalmodel.Minordeviationswereobservedinthe pressuremagnitudes(seeFig.12d).Thesedeviationswereobserved,ex- ceptforthelowestpressuremagnitude,inboththeincomingandthe reflectedoverpressure.Theincomingpressuresandthefirstpartofthe reflectedoverpressurewereslightlyunderestimated,whilethereflected overpressuretendstobeslightlyoverestimatedthroughouttheremain- ingpartofthepressurehistories(Fig.12d).Itishoweverchallenging toconcludeontheinfluenceof theseminordeviations regardingthe dynamicresponseofthethinsteelplates. ThisisbecauseSensor1is locatedupstreamofthetestspecimen(seeFig.1a),sinceitwouldbe difficulttomountpressuresensorsonthethinplatewithoutalteringits structuralcharacteristics.Thus,Sensor1doesnotmeasurethepressure onthesteelplatesbutratherthepressureintheroofoftheshocktube and24.5cmupstreamofthethinsteelplates.
Sincetheseareminordeviationsifoneconsidersthesophisticationof thenumericalmethodologynecessarytoaccuratelypredictthepressure wavesoccurringinthesetypesofshocktubetests,itcanbeconcluded thattheoverallperformanceofthefluidsub-domainisacceptable.Itis importanttoemphasizethatthereisconsiderablecomplexityinthese simulations,whereoneofthemainchallengesistomodelthediaphragm failureprocessanditsinfluenceontheresultingwavepatternsinsidethe shocktube.Furtherimprovementsofthenumericalaccuracyinpredict- ingtheblastwaveformationintheSSTFwillneedtofocusevenmore onthemodellingofthediaphragmfailureprocess(seeFig.5).Thisas- pecthasalreadybeenidentifiedinarecentstudybytheauthors[51],
Fig. 12. Comparison of experimental mea- surements andcorrespondingnumericalpre- dictionsinthefullycoupledsimulations:(a) Mid-pointdeflections,(b)mid-pointvelocities, (c)pressuresmeasuredatSensor1and(d)the samedataasFig.12cwherethetimeaxisis more focused on thereflected overpressure.
NotethatthecurvesinFig.12c-dareshiftedin timeforimprovedreadability.Eachcurvewas right-shifted3mswithrespecttotheprevious one.
Fig.13. Deformationprofilesat0%,25%,50%,75%and100%ofmaximum deflectionforthecoupledapproachandexperimentaldataintestD35.The deformationprofilesareextractedfromthecentrealongthex-axis.
butitisleftasfurtherworksincethelevelofaccuracyalreadyreached issufficient toevaluateFSIeffects.Detailed investigationsof thedi- aphragmfailureprocesscanthereforebeconsideredbeyondthescope ofthepresentstudy.ThisstudyfocusesontheFSIeffectsduringthe dynamicresponseoftheplatesusingapurelynumericalapproach.Ex- perimentalvalidationisonlyusedtoevaluatethepredictivecapabilities ofthenumericalmethodology.Thealreadyverygoodagreementwith experimentalobservationsinFigs.12and13providesconfidenceinthe useof thepresentnumericalmodelandinthefactthatthenumeri- calstudiesonFSIcanbecarriedoutnumerically,bothinaqualitative andquantitativemanner.Hence,thenumericalmethodologypresented
hereincanbeusedtoobtainmoreinsightintotheunderlyingphysics observedintheexperiments.
5. Concludingremarks
Thepresentstudyinvestigates FSIeffectsduringthedynamic re- sponseofblast-loadedsteelplates.Sucheffectswerestudiedbycom- paringthenumericalpredictionsoftheuncoupledandcoupledFSIap- proach.PurelyEuleriansimulationswereusedtogeneratetheloading intheuncoupledapproach.Specialfocuswasplacedontheinfluenceof FSIonthemid-pointdeflectionsandmid-pointvelocitiesintheplates, whereexperimentaldataservedasabackdroptoevaluatetheaccuracy ofthenumericalsimulations.Themainconclusionsfromthestudyare asfollows.
• Fluid-structureinteraction(FSI)takesplaceifthesteelplatemoves ordeformsduringthedurationoftheblastload.
• TheinfluenceofFSIeffectsisquantifiedintermsofthedeflections, thepressuresandthesaturatedimpulse.Asexpected,theuncou- pledapproachprovidesconservativepredictionsforthedynamicre- sponseintheplates,i.e.,itoverestimatestheplatedeflection,dueto theinherentassumptionthatthepressureisunalteredbytheplate deformation.Thecleartrendisthathigherblastpressuremagnitudes resultinincreasedFSIeffectsinthefullycoupledsimulations,result- inginreduceddeflectionsandvelocitiesintheplate.Itwasobserved thatthemid-pointdeflectionswerereducedby4–14%,depending ontheblastintensity,whenconsideringFSI.
• Fullycoupledsimulationsshowedthatthedynamicresponseofthe steelplateintroducesanon-uniformspatialandtemporaldistribu- tionofthepressureneartheplate.Thefactthattheinducedvelocity intheplatetendstoreducethepressurewasconfirmed,inaccor- dancewithpreviousstudiesintheliterature.However,theobserved successiveincrease inpressuredue tothedeformedshape of the platewasunexpected.Itwasinterestingtonotethatthepressure
