metals Effect of nodal mass on macroscopic mechanical properties of nanoporous International Journal of Mechanical Sciences

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International Journal of Mechanical Sciences

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Effect of nodal mass on macroscopic mechanical properties of nanoporous metals

J. Jiao

, N. Huber

aInstitute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht, Germany

bInstitute of Materials Physics and Technology, Hamburg University of Technology, Germany

a r t i c le i n f o

Keywords:

Nanoporous metal Finite element method Nodal mass Macroscopic response Scaling laws Size dependent strength

a b s t r a ct

Thecurrentworkinvestigatestheeffectofthenodalmassonthemacroscopicmechanicalbehaviorofnanoporous metalsusingtheFiniteElementMethod.Anodalcorrectedbeammodelingconceptisintroducedthatallows localincorporationoftheeffectiveelastoplasticmechanicalbehaviorofthenodalmassinthenodalareaofa representativevolumeelement(RVE).ThecalibrationtothecorrespondingFiniteElementsolidmodelisachieved byintegratingadditionalgeometryandmaterialparameterstotheso-callednodalareasinthebeammodel.With thistechniqueanexcellentpredictioncanbeachievedoveralargerangeofdeformationfordifferenttypes ofRVEs.Fromtheresultsofthenodalcorrectedbeammodel,modifiedleadingconstantsaredeterminedin thescalinglawsforYoung’smodulusandyieldstrength.Theeffectofthenodalcorrectionisalsostudiedwith respecttovariousrandomizationlevels.Finally,theligamentsizedependentstrengthisanalyzedbyapplyingthe proposedmodeltoexperimentaldata.Itcouldbeshownthatthenodalcorrectionimprovestheoverallagreement withliteraturedata,particularlyforsuchdatapointsthatarerelatedtosampleswithahighsolidfraction.

© 2017TheAuthors.PublishedbyElsevierLtd.

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1. Introduction

Nanoporousgold(NPG)madeby de-alloyingcan beproducedas macroscopicobjectsthatexhibitabi-continuousnetworkofnanoscale poresandsolid‘ligaments’ whichareconnectedinnodes.Thesolidfrac- tion𝜑oftheporousbodyisapprox.30%[1–4].𝜑isusedasthemajor parameterinseveraltheoreticalmodelsforpredictingthemacroscopic mechanicalbehavioroftheporousmaterials[5–9].TheGibson-Ashby model[10], asthemostcommonlyusedone amongthese modelsis reportedtosignificantlyoverestimatethemacroscopicmechanicalre- sponseofnanoporousmetals[11–14].Theoverestimationindicatesthat themassutilizationfordeformationinsuchamaterialisnotasefficient asassumedbytheGibson-Ashbystructuralmodelforopenporefoams.

Inawider spectrumof attemptsforunderstanding theextraordi- narymechanicalresponsesofnanoporousmetals,extensivemodeling approacheshavebeenconducted. Atomisticandmoleculardynamics simulationshavebeenimplementedtoinvestigatethedeformationbe- haviorundertensionandcompression[6,15–17].Itwasfoundthatthe surfacestresshasasubstantialimpactonthetension/compressionasym- metry,anomalouscomplianceandearlyyield.Thesimulationsreported in[16]revealedsignificantstackingfaultformationanddislocationac- cumulationwithinthenanosizedligaments,confirmingtheexistence

∗Corresponding author.

E-mail address: jingsi.jiao@hzg.de (J. Jiao).

ofsubstantialworkhardeningunderplasticdeformationassuggested in[8].Further modelingworkonthemicrostructuralandcontinuum level[18,19]wasconductedtoexploretheorigin oftheunusuallow Poisson’sratioobservedduringmacroscopiccompressionofnanoporous goldsamples.ItwasfoundthatontheonehandtheelasticPossion’sra- tio isindependentoftheligamentsizebutdecreaseswithincreasing degreeofrandomizationthroughanincreasingpercentageoftorsionof theligaments[18].Ontheotherhand,theplasticPoisson’sratioshowed astrongdependencyontheligamentsize,whichcouldbesuccessfully reproducedwiththeDeshpande–Fleckmodel[19].

Otherworks[20–23]thatstudiedsurfaceelasticityorsurfacebound- aryconditionshavebeenconductedaimingtoexplainthemacroscopic mechanicalbehaviorsofthismaterialfromamicroscopicpointofview, withaparticularemphasisonsizeeffects.Veryrecently,thesignature ofthesurfaceenergywasstudiedbycombiningmacroscopiccompres- sionexperimentswithaFiniteElementbeammodelofarandomized diamondstructure,enriched bycoaxialthin-walledtubularelements for modelinga switchablesurfacestress[24].Theresults showedin conjunctionwiththeexperimentalfindingsthat,contrarytotheelastic Poisson’sratio,theplasticPoisson’sratiorespondsstronglytoelectri- calsurfacemodulation.Thisbehaviorwasidentifiedasthesignature ofasurface-inducedtension-compressionasymmetryoftheflowstress

https://doi.org/10.1016/j.ijmecsci.2017.10.011

Received 23 June 2017; Received in revised form 20 September 2017; Accepted 7 October 2017 Available online 9 October 2017

0020-7403/© 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

sizeofthenodesconnectingtheligamentsisintroducedintheserefined models.Theextramassofthenodesisinvestigatedregardingthecal- culationofthesolidfraction𝜑andthemechanicalresponseoftheunit cell.Itisconcludedthatthenodalmassshouldbecountedasanother importantfactorthatisassociatedwiththerelationbetweenaspectratio oftheligaments,solidfractionandmacroscopicmechanicalproperties.

Morespecifically,itwasindicatedthatthediscrepancybetweentheex- perimentalmeasurementsandtheGibson-Ashbymodelpredictionisdue tothatthereisamassivemassaccumulatedatthenodes,whichdoes notcontributetotheelasticdeformationoftheunitcell[9,12,25].

Theeffectfromthenodesontheelastic-plasticdeformationwasin- vestigatedbycomparingresultsfromfiniteelementsimulationsfora finiteelementsolidmodelwithsphericalnodesandafiniteelement beammodel,thatreducesthenodestoacouplingconstraintofthecon- nectedbeamelements,basedonthediamondunitcell[8].Itwasfound thattheelasticstiffeningcausedbythenodesispartiallycompensated bytheeffectofincreasingsheardeformationforthickligaments.Con- cerningtheplasticdeformationbehavior,theshorteningoftheligament lengthduetothenodalmasswasidentifiedasanimportanteffectthat wasincorporatedasacorrectionfactorintheproposedscalinglaws.

Thus,forthescalinglawofthemacroscopicmechanicalbehavior,the structuralarrangementoftheligamentsisnottheonlyinfluentialpa- rameter;thedifferentapproachesformodelingthenodesintermsof theircontributiontothesolidfractionaswellaselasticandplasticde- formationbehaviorwillalsoleadtodifferentscalinglawscomparedto theGibson–Ashbymodel.

Thestudiesonnanoporousmetalscannotberestrictedtoasimpli- fied,geometricallyperfectunitcellstructure,duetothattherealstruc- tureofnanoporousmetalsisaspatialnetworkstructurewithcomplex topologicalandmorphologicalcharacteristics[7,13,26,27].Inorderto investigatethestructuralparametersandobtainamorerealisticdefor- mationoftheligaments,highlygeneralizedbeamelementsareused, whichofferexcellentcomputationalefficiencyevenwiththousandsof ligamentsmodeledinarepresentativevolumeelement(RVE).Basedon thediamondlatticestructurethatwasestablishedintheworkof[8]and furtherrefinedby[29],theeffectsofligamentshapevariationandran- domizationonthevariousdeformationmodesofbending,torsion,and tension/compressionwasinvestigated[18].Itcouldbeshown,thatthe beamelementB31inABAQUS[28]iscapableofcapturingallfunda- mentaldeformationsforeventhick beamswithaligamentradiusto lengthratio,r/l,upto0.5correspondingtoasolidfractionof69%.So faroneremainingdrawbackoftheRVEbuiltfrombeamelementsisthe unsolvedquestionofhowtomodelthemassinthenodalareas,because theligamentsinthemodelareconjugatedatvirtualnodes.Thisleadsto alargercomplianceandlowerstrengthofthebeammodel.Inthestudy of[29],acorrectionofthemacroscopicyieldstrengthwasconsidered byincreasingtheyieldstrengthofthesolidphasebyafactorthatwas derivedfromtheavailableleverforbendingoftheligamentwhichis shortenedbythenodalmass,basedonthecorrectionderivedin[8]. Itishoweverunclear,howaccurateandhowgeneralthisapproachis, particularlywhentheRVEhasincreasingcomplexity throughadding randomizationorvariationsofligamentshape.

lyzedandcomparedtopreviousstudies.Gibson-Ashbyscalinglawsfor elasticityandplasticitywithmodifiedleadingconstantsareproposed thatincorporatetheeffectofthenodalmassonthemacroscopicme- chanicalresponse.Theroleofthenodalmassonthemacroscopicme- chanicalresponseisalsoanalyzedwithrespecttovariousrandomization levels.Finally,theligamentsizedependentstrengthofnanoporousgold is determinedbyapplyingtheproposedmodeltoexperimentaldata.

Theresultsarediscussedinthelightofpreviousstudies[29]. 2. Extensionofthebeammodel

ThegoalofthissectionistodevelopaRVEbuiltwithbeamelements, wherethemechanicsoftheconnectingnodesisphysicallyandlocally includedineachofthenodesofthebeamstructure.Itshallserveforpre- dictingtheelastic-plasticmacroscopicresponseofananoporousmetal withhighcomputationalefficiency.Thisapproachwouldallowelimi- natingthephenomenologicalcorrectionfactorthathaslimitedaccuracy anddoesnotallowconsideringeffectsoflocalstructuralvariationsin thevicinityofthenodes.

Inwhatfollows,abeamnodalcorrectionapproachwillbeproposed thatisbasedonatetrahedronstructure– thebuildingblockofthedi- amondstructure– thatallowsstudyingthemechanicalresponseona moreadvancedlevelcomparedtoasingleligamentRVE.Thetetrahe- dronstructureservesforadjustingtheelastic-plasticdeformationbe- haviorofthebeammodelinrelationtoasolidmodelofsamegeometry againstbending.Theapproachwillbevalidatedwithrespecttotorsion onthelevelofthetetrahedronstructureand,inasecondstep,forlarger RVEsthatarecomparedtosolidRVEs.

2.1. Tetrahedralbuildingblock

Atetrahedronstructureconsistingoffourligamentsissketchedin Fig.1.Itserves asthefundamentalbuildingblockfortheRVE,pro- posedby[8].Fig.1ashowsthestructuremodeledwithsolidelements.

Thegeometryconsistsof thenodalareain formof asphericalmass connectedwithfourligaments,of whichonlyhalfof theligamentis modeled.Fig.1bdemonstratestheequivalentstructurethatismodeled with beamelements. Theligamentsare connectedbytheir common nodevialinking alldisplacementandrotationaldegreesoffreedom.

Asthenatureofthebeammodel,thenodalareacontainsintersecting volumesoftheligaments.Thus,thesolidfractionhastobecalculated independentofthevolumeoftheelementsinthefiniteelementmodel basedontheinitialgeometryshowninFig.1a.Whilethecalculation ofthesolidfractioncanbeeasilycarriedoutfortheperfectlyordered diamondstructure[8]andalsofortherandomizeddiamondstructure [29], thecorrectmechanicalresponserequiresamodification of the beammodel.

Asbendingisthedominantdeformationmechanisminnanoporous metals,thecalibrationofthemodelparameterswillbecarriedoutfor suchadeformation.Thisisachievedbyapplyingatransversedisplace- mentwattheendofthetopligamentwhilefixingtheotherthreelig- amentendsinspace(showninFig.1aandb).Thefollowingapproach

Fig. 1. Geometry of the tetrahedron building block for (a) solid model and (b) beam model.

Fig. 2. Parameters defining the geometry of the ligament-nodal structure (a) solid model and (b) beam model.

Table 1

Geometry parameters and resulting solid fractions for simulations with variation of r / l and c R.

r / l = 0.19 r / l = 0.25 r / l = 0.31

c _R= 1 c _R= 1.05 c _R= 1.1 c _R= 1 c _R= 1.05 c _R= 1.1 c _R= 1 c _R= 1.05 c _R= 1.1 𝜑 = 0.126 𝜑 = 0.127 𝜑 = 0.129 𝜑 = 0.207 𝜑 = 0.210 𝜑 = 0.214 𝜑 = 0.301 𝜑 = 0.307 𝜑 = 0.313

aimsatadjustinggeometricalandmaterialparametersofthebeamele- mentsinthenodalarea,suchthatthemechanicalresponseofthebeam modelisequivalenttothatofthesolidmodel.

Forintroducing thegeometricalparametersofthemodifiedbeam model,oneligamentwithitsconnectingnodeisschematicallyshownin Fig.2aandb.ForthesolidmodelinFig.2a,theparametersRandl_n arethenodalradiusandthelengthbetweennodalcenterandtheendof theligamentattachedtothenode,respectively.Inthenodalcorrected beammodel,shown inFig. 2b,𝑟^∗_𝑛 and𝑙^∗_𝑛 areadjustablegeometrical parametersdescribingthelargercross-sectionofthebeamelementsin thenodalareaandthelengthbetweenthenodecenterandtheendofthe ligament,respectively.Theligamentradiusrandligamentlengthlthat isdefinedasthedistancebetweenthetwonodecenters,arecommon parametersforthesolidmodelandthebeammodel.Anotherparameter c_R isintroduced,whichis aconstantgoverningtheadjustablenodal massbylinkingtheligamentradiusrandnodalradiusRintheformof 𝑅=√

3∕2𝑐𝑅⋅𝑟[8].

AllthecalculationsinthestudyareconductedwiththeFEAcode ABAQUS/Implicit[28].Thegeometryofthesolidandthebeammodel ismeshedwithR3D3andB31elements.InTable1threer/lratiosand c_Rvaluesarelistedthatresultinninemodelrealizations.Therangeof r/lisselectedbasedontherangefromthemorphologicalstudyforthe ligamentsofNPGintheworkof[9],whichincludesthetypicalrangeof thesolidfractionofNPGfrom0.25to0.3[8,29].Avalueofc_R=1leads totheminimumpossible sizeof theconnectingnodewherethefour ligamentstouch,whilec_R=1.1isreportedtorepresentamorerealistic estimationofthenodesizeinnanoporousmetals[8].

Toapplythesameloadingconditiontothesolidmodelastothe beammodel,thesolidligamentcrosssectionalsurface(Fig.1a)iscou- pledtoarigidplatewitha‘Coupling’ constraintthathasa‘Continuum distribution’.Theloadingcanbethereforeassignedtothecenterpoint of therigidplateanddistributedtothewholesurface[28],ensuring themaximumlevelofthesimilaritybetweenthesolidandbeammod- elswithrespecttotheloadapplication[18].

Thematerialbehaviorofthesolidfractionisassumedtobeelastic- perfectlyplasticwithaYoung’smodulusE_Sof81GPaandaPoisson’s ratio𝜈of0.42.Theyieldstrength𝜎y,Sis500MPaandplasticdeforma- tionevolveswithoutworkhardening[8,29].

2.2. Calibrationofthenodalbeamelements

Asmentionedintheprevioussection,thecalibrationofthebeam modelisconductedwiththereferencetothemechanicalresponseofthe solidmodelunderbendingdeformation.Fig.3illustratesthevonMises stressdistributionforasolidtetrahedronstructure(r/l=0.25,c_R=1.1) underthetransverseloadingdisplacementwaccordingtoFig.1athat leadstoplasticbendingandshearingofthetopligament,butalsoofthe upperpartoftheconnectingnodeanditssurroundingligaments.

Thecorrectionforthemechanicalresponseofthebeamtetrahedron structure iscarried outbyfirstlyadjustingitsadditionalgeometrical parameters𝑟^∗_𝑛 and𝑙^∗_𝑛.Tothatend,thestiffnessofthetetrahedron(for bothofthebeamandsolidmodels),k=F/w,iscomputedfromthefirst loadingincrementintheelasticregime,suchthatthestructureisun- dergoingonlysmallelasticdeformation.ThestrengthFisreadasthe reactionforceofthetetrahedronstructureintheplasticregimewhen

Fig. 3. Mises stress distribution of solid model with r / l = 0.25, c R= 1.1; top ligament and upper part of the nodal mass and lower ligaments are plastically deformed.

w/l=0.017.Thisdeformationvalueleadstoamacroscopicdeformation withintheplasticregimeoftheforcedisplacementcurve,whichissuf- ficientlyfarfromtheelastic-plastictransition.Thedatak_solidandF_solid, determinedfromthesolidmodel,areusedasreferencesforthecali- brationofthebeammodel’selasticandplasticresponse,represented byk_beamandF_beam,respectively.Fig.4ashowstheratiosofk_beam/k_solid (blackcrosses)andF_beam/F_solid(redcircles)forthecaseofr/l=0.19and c_R=1.1.Theanalyzisisconductedwithsystematicallyvaryingthepa- rameters𝑟^∗_𝑛and𝑙^∗_𝑛;anditisdoneforallgeometrieslistedinTable1.The bestfitisfoundwherestiffnessandstrengthresultsaresimultaneously closesttothevalueof1,representedbytheyellowplaneinFig.4a.

ItcanbeseenfromFig.4athatthestiffnessk_beamsmoothlyincreases with𝑟^∗_𝑛.Incontrasttothat,thestrengthratiofirstincreasesinthesame waywith𝑟^∗_𝑛butthenarrivesataplateau.Thisisbecauseplasticyielding wouldalwaysinitiateattheendoftheligamentoncethenodalregion issufficientlystrong.Untilthen,thenodalregionalsocontributestothe plasticdeformation,asshowninFig.3.Themagnitudeoftheplateau isrelatedtotheleverlengthoftheligamentthatcanbeobtainedinthe formof𝑙−𝑙^∗_𝑛.Alargervaluefor𝑙^∗_𝑛resultsinsmallerleverlengthwith theconsequencefortheyieldingtorequirealargerexternalforce,which againleadstoanincreaseoftheplateauvalueasshowninFig.4a.

valueF_beam/F_solid ≥1representedbytheyellowplaneinFig.4a.

Thisisdonebytuningthenumberofbeamelementsincludedinthe nodalarea,assignedwiththenodalradius,𝑟^∗_𝑛,whiletheremaining elementskeeptheradiusoftheligament,r.

b) 𝑟^∗_𝑛issecondlyadjustedforthebeammodeltoreproducethecorrect strengthofthesolidmodelsuchthatF_beam/F_solid =1.

c) Thefinalcalibrationof𝐸^∗_𝑛willresolvetheremainingcalibrationof thestiffnessk_beam/k_solid =1withoutaffectingthestrengththatwas calibratedbefore.

Accordingtothestepslistedabove,forthecaseshowninFig.4a, firstly𝑙^∗_𝑛∕𝑙_𝑛=2.23isdetermined,whichisfollowedbythedetermination of𝑟^∗_𝑛∕𝑟=1.4.Thefittingiscompletedbyadjusting𝐸_𝑛^∗to𝐸_𝑛^∗∕𝐸𝑠=0.75 and, as illustrated in Fig. 4b, a 99% agreement is simultaneously reachedforthestiffness andstrengthratio, markedbytherectangle inFig.4b.Itshouldnotedthatanexactmatchinstepa)isnorrequired neitherdesired,becausewewanttousetheinitialdiscretizationofthe ligamenttoformtheelementsinthenodalareasintherandomizedRVEs structures.Otherapproacheswithvariableelementlengtharealsopos- sible,butwouldneedaremeshingofnodalareaaswellastheremaining ligament.

AllninegeometrieslistedinTable1wereanalyzedfollowingthe stepsintroducedabove.Anoverallfittingaccuracyof98%isachieved simultaneouslyforstiffnessandstrength.TheresultsshowninFig.5a–d arepresentedinformoftheratiosoftheadjustableparametersofthe nodalcorrected beammodeltothecorresponding parametersofthe solidmodel,𝑙^∗_𝑛∕𝑙_𝑛,𝑙^∗_𝑛∕𝑙,𝑟^∗_𝑛∕𝑟,and𝐸_𝑛^∗∕𝐸_𝑠respectively.Theseratioscan beinterpretedasindicatorsforthegeometricaldifferenceofthenodal

Fig. 4. Comparison of the mechanical response obtained for the beam and solid model for the case r / l = 0.19 and c _R= 1.1. (a) Scan with variation of geometrical parameters 𝑟 ^∗_𝑛and 𝑙 ^∗_𝑛; best agreement is achieved along the curve 𝑙 ^∗_𝑛∕ 𝑙 _𝑛= 2.23 at 𝑟 ^∗_𝑛∕ 𝑟 = 1.4; (b) Final fitting with 99% accuracy of stiffness and strength by additionally adjusting the material parameter 𝐸 ^∗_𝑛to 𝐸 _𝑛^∗∕ 𝐸= 0.75.

Fig. 5. Identified values for the adjustable parameters of the nodal beam elements providing an 98% agreement of the tetrahedron beam model with the solid model with respect to stiffness and strength for the cases listed Table 1 (a) 𝑙 _𝑛^∗∕ 𝑙 _𝑛; (b) 𝑙 ^∗_𝑛∕ 𝑙; (c) 𝑟 ^∗_𝑛∕ 𝑟 ; and (d) 𝐸 _𝑛^∗∕ 𝐸 _𝑠ratio.

areamodeledbybeamelementssuchthatitpredictsthemacroscopicre- sponseofasolidmodel.Fig.5ashowsthatthevalueof𝑙^∗_𝑛∕𝑙_𝑛isstrongly dependentonther/lratioandc_R.AsitisfurthershowninFig.5b,the magnitudeof𝑙^∗_𝑛∕𝑙(liskeptconstant)foragivenr/ldoesnotdependon c_R.Thisshowsthatthedependencyof𝑙_𝑛^∗∕𝑙_𝑛 onc_RshowninFig.5ais causedbythecalculationofthel_nvaluesasfunctionofc_R.Thedecrease of𝑙^∗_𝑛∕𝑙𝑛 from2±0.2to1±0.1inFig.5asuggeststhatthesizeofthe nodalareaisconvergingwithincreasingligamentsize.Inotherwords, thebeammodel’splasticbehaviorbasedonthesamegeometryiscloser tothatofthesolidmodelforlargerligamentsizes(thisisfurtherdis- cussedinsect.4.1).Thesamelineofargumentsappliestotheratio𝑟^∗_𝑛∕𝑟, seeFig.5c.

Thethirdadjustablematerialparameter,𝐸_𝑛^∗,controlstheelasticbe- haviorof the nodalarea.Fig. 5ddemonstrates a generaltrend that theratioof𝐸^∗_𝑛∕𝐸_𝑠increasesfrom0.75forr/l=0.19to2.5±0.04for r/l=0.31.Therequiredcorrectionof theeffectivenodalstiffness in- creasesprogressivelywithincreasingr/l ratioand,atthesametime, itshowsanincreasingsensitivitywithregardtothenodalextension, representedbyc_R.Thegeneraltrendfromavaluebelow1toincreas- ingvaluessimplyresultsfromthediscrepancybetweenthestiffnessand strengthratiothathasbeendiscussedincontextofFig.4a.Thatratio followstheoppositeandhastobecompensatedbyanincreasinglocal Young’smodulusassignedtothenodalbeamelements.

Eqs(1)–(3)arethefittingfunctionsthataregeneratedbasedonthe datesetsfromFig.5(a),(c)and(d),respectively.Thefittingconstants arelistedinTable2.

𝑙^∗_𝑛∕𝑙_𝑛=𝑎0+𝑎1⋅𝑟∕𝑙 (1)

𝑟^∗_𝑛∕𝑟𝑛=𝑎0+𝑎1⋅𝑟∕𝑙 (2)

𝐸_𝑛^∗∕𝐸=𝑎0+𝑎1⋅exp( 𝑎2⋅𝑟∕𝑙)

(3)

2.3. Modelvalidation

Itwasreportedintheworkof[18]thatbesidesbendingasthemajor deformationmechanism,torsionrepresentsanotherfundamentaldefor- mationthatiscausedbythespatiallycurvedligamentsandshouldnot beneglected.Thetorqueintheligamentsoccupiesapprox.20%ofliga- mentloadingundermacroscopiccompressingofnanoporousgold.The validationoftheproposedmodelisthereforefirstlycomparingthenodal correctedbeamtetrahedronmodeltothesolidmodel,seeFig.1,under externaltorsionloading.

Fig.6showsthecomparisonbetweensolidmodel(SM)andbeam tetrahedronundertorsion.ThetorqueM_thasbeennormalizedwithEr³.

Fig. 6. The mechanical responses of solid model (SM), beam model (BM) and nodal cor- rected beam model (NCBM) for the tetrahedron structure under external torsion loading for the case of r / l = 0.25, c _R= 1.1.

Theoverallfittingofthenodalcorrectedbeammodel(NCBM)iscom- parabletothatofthebeammodel(BM),reachingabeamtosolidstiff- nessratioof130%andanapprox.80%overallaccuracyfortheplastic deformation.Theperformancefortorsionwasfoundtobeconsistent acrossthewholerangeofr/lstudiedinthecurrentwork.Asinrandom- izedRVEsapprox.80%oftheligamentdeformationisoriginatingfrom bendingasthegoverningdeformationmechanismandapprox.20%is relatedtotorsion[18],itisreasonabletoassumethattheoverallerror inthemacroscopicresponseoftheRVEiskeptwithin10%.

Asdiscussedintheintroduction,theRVEbeammodelprovidesan idealaccessforstudyingthemechanicsofnanoporousmaterials.Incon- trasttotheanalyzesbasedonabuildingblock,anRVEhowevercon- tainsthousandsofligamentsinterconnectedinanetworkstructure.Such anetworkstructureis muchmorerepresentativeoftherealmaterial intermsofstructuralfeatures,localloadingoftheligamentsaswell asobservabledeformationmechanisms.Thecurrentnodalcorrection methodisthereforeintegratedintotheRVEbeammodelusingthedi- amondstructure,developedintheworkof[8].ThreeRVEshavebeen generatedassolidandbeammodelsforc_R=1.1andligamentaspect ratiosofr/l=0.19,0.25,and0.31.InFig.7aandbtheRVEsareshown forr/l=0.25forthesolidandthebeammodel(inrenderingmode), respectively.TheRVEsareloadedwiththesamecompressiondeforma- tionof25%engineeringstrainonthetopsurface.Althoughthestructure isperfectlyordered,itrepresentsafurtherstepofvalidation,because themacroscopiccompressionoftheRVEtranslatesintoadifferentlocal loaddistributionanddeformationinthetetrahedronsasappliedduring thecalibrationofthenodalcorrectedelements.

Theresulting agreementfor themacroscopic elasticmodulusbe- tweenthebeamandsolidRVEis100±5%forallthree cases.Thus, thenodalcorrectedbeammodeliscapableofpredictingtheelasticme- chanicalresponseoftheRVEequivalentlytothesolidmodel.

Becauseanelastic-perfectlyplastic materialmodelis usedso far, earlyconvergenceproblemshavebeenfacedinallbeamRVEsimula- tions,possiblyowingtobuckling.Therefore,amore realisticelastic- linearplasticconstitutivelawisemployedtothesimulationsofRVEs forbothofthebeamandsolidmodels,usingthesameelasticproper-

examiningthegeneralityofthenodalcorrectedbeammodelingconcept inpresenceofworkhardening.

Fig.8ademonstratestheresultingmacroscopicstress-strainresponse fromthenodalcorrectedbeamRVEandthesolidmodelRVEfordif- ferentr/lratiosincomparisonwiththeresultsfromtheoriginalbeam model.Thenodalcorrectedbeammodel(NCBM)significantlyimproves thepredictionsofthemacroscopicmechanicalresponsebyeliminating thesystematicunderestimationofstiffnessandstrengthoftheoriginal beammodel(BM).Furthermore,averygoodagreementwiththesolid model(SM)isachievedoverthewholestrainrange.Fig.8bshowsthe ratioof theresultingreaction forceobtained forthenodalcorrected beamRVEandthesolidmodelRVEagainstthemacroscopiccompres- sionstrain,𝜀eng.Theoverallerroroftheelastic-plasticresponseofthe nodalcorrectedbeamRVEiswithinabout10%forallsolidfractions(i.e.

allconsideredr/lratios).Morespecifically,theaccuracyis101%±5%

forr/l=0.19and0.31,and103%±7%forr/l=0.25.Itcanbeconcluded thatthenodalcorrected beammodelprovidesasufficientlyaccurate elastic-plasticmechanicalresponsethathasthesamepredictivequality ofacorrespondingsolidmodel.Atthesametimeitcostsmuchlesscom- putationtimeandprovidesmuchmoredegreesoffreedomconcerning theintroductionofarandomstructure[8,29]aswell asvariationof ligamentshapesintheRVE[18].

3. Discussionofscalinglaws

InthissectionwewillapplythenodalcorrectedbeamRVEproposed andvalidatedintheprevioussectionsforanalyzingtheimpactofthe nodalcorrectiononthescalinglawsforYoung’smodulusandstrength.

Theresultingcurvesarecomparedtoscalinglawsfromliterature.For thenodalcorrectedbeamRVE,thesolidfraction𝜑isderivedfromthe originalgeometrythatcorrespondstothesolidRVE.

Fig. 9a andbrepresent the scalingbehaviorof the macroscopic Young’smodulusandstrengththat arecomputed fromtheRVEnor- malizedtothecorrespondingmaterialpropertiesofthesolidfractionin theformE/E_S,and𝜎y/𝜎y,S,respectively.Thecurvesareshownincom- parisonwithscalinglawssuggestedinpreviousworks[8,29],aswellas theoriginalGibson-Ashbyscalinglawsthatserveasthecommonrefer- enceforallmodels.Foropenporefoamsthatimplybendingasmajor deformationmechanism,theleadingconstantsinEqs.(4)and(5)are C_E=1andC_𝜎=0.3respectively[10].

𝐸∕𝐸𝑆=𝐶𝐸𝜑² (4)

𝜎𝑦∕𝜎𝑦,𝑆=𝐶𝜎𝜑^3∕2 (5)

InFig.9a,thescalingbehavioroftheYoung’smodulus,E/E_S,calcu- latedfromthedifferentRVEsisplottedagainstthesolidfraction𝜑.The different𝜑valuesaretheresultsofvaryingr/lratioswith0.19,0.25and 0.31,giveninTable1.TheblacksolidlinecorrespondstoEq.(11)in [8]whichdescribesthescalinglawoftheRVEbeammodelundercon- siderationoftheeffectofsheardeformationforthickbeams,i.e.itisthe approximationofthelowerboundforthediamondRVE.Thediamond

Fig. 7. RVEs for r / l = 0.25 and c R= 1.1 (a) solid model; (b) nodal corrected beam model with the same effective elastoplastic response.

Fig. 8. (a) Macroscopic stress-strain response for RVEs built as solid model, beam model, and nodal corrected beam model for various r / l ratios; (b) RVE reaction force ratios F beam/ F solid

for different r / l values covering the whole range of elastic and elastic-plastic deformation.

Fig. 9. Comparison between the nodal corrected beam model and previous studies for (a) Young’s modulus; (b) yield strength. RVE-SM, RVE-NCBM, and RVE-BM denote the results from the RVE solid model (this work, Fig. 7 a), the RVE nodal corrected beam model (this work, Fig. 7 b) and RVE beam model [29] , respectively. All results are for perfectly ordered diamond structure ( A = 0).

studiesshowedthatrandomizationofthestructureleadstoafurtherin- creaseincompliance[8,29].Inrecentstudies[14,30],experimentally constructedscalingrelationsaredevelopedforthemacroscopicYoung’s modulusofnanoporousgold,whichsuggestthepowerinEq.(4)being 2.5or2.8withC_E=1or0.86.However,theincorporationofthenodal massinthisworkdoesnotleadtoanexponentdifferentto2thatac- cordingto[10]representsthehighestpowerofalltypesofdeformation (tension,compression,shear,bending).Atthispointthequestionre- mainsunsolvedhowhighervaluesintheexponent,asobservedfrom experiments,canbeexplainedbyastructuralmodelsuchastheRVE presentedinthiswork.

Fig.9bshowstheresultsforthescalingbehavioroftheyieldstrength 𝜎y,normalizedbytheyieldstrengthofthesolidphase𝜎y,S.Again,the analyticalsolutionforthebeammodelproposedasEq.(9)in[8]pro- videsalowerlimitthatcorrespondstothenumericalresultsfromthe beammodel(RVE-BM).ItcanbealsoobservedthattheGibson-Ashby modelwithC_𝜎=0.3resultsinagoodagreementwiththatscalingrela- tion.

Theanalyticalnodalcorrection(blacksolid line)proposedin the workof[8]asEq.(10)formstheupperlimitwhichagreeswellwith theresultsfromtheworkof[29]representedbyrhombussymbolsin Fig.9b.In[8]themacroscopicstrengthfortheRVEbeammodelwas correctedbyreducingtheavailableleverforbendingoftheligament bytheradiusofthenodalmass.In[29]thiseffectwastranslatedinto thebeammodelbyincreasingtheyieldstrengthofthesolidfraction accordingtothisstructuralstrengtheningeffect. Thelatter approach hastheadvantage,thatitdoesnotaffectthestiffnessofthenetwork structure.Forbothoftheworks,thenodalmassistakenaccountasa shorteningoftheeffectivebendingleveroftheligamentsandassuming thatyieldoccursattheedgeformedbytheligamentandthenode.The resultsshowninFig.9bleadtotheconclusionthatthistypeofcorrec- tionprovidesreasonableresultsforsolidfractions𝜑≤0.15.Ithowever significantlyoverestimatestheeffectofthenodalmassonthemacro- scopicstrengthfor𝜑>0.15,whichiswherethenanoporousmetalsare located.Inviewoftheseinsights,wewillrevisittheanalysisof[8]and [29]withsupportofthenodalcorrectedbeammodelinSect.4.2.

TheresultsofthenodalcorrectedbeamRVE-NCBM(blackcircles) initiallyfollowthesolutionfortheupperlimitbutthendeviatebycon- tinuinginamuchlessprogressivewayfor𝜑>0.15.Thisisbecause thattheupperlimitisconstructedundertheassumptionthattheplas- ticdeformationwillalwayshappenattheendoftheligamentthatis analyticallyshortenedforthepresenceofanode.Thismeansthatthe nodalareaforthisupperlimitissettobeinfinitelystrong,e.g.noplastic deformationcanhappenwithinthenodalarea.However,thevisualin- vestigationonthelocalplasticdeformationofthenodalcorrectedbeam modelandsolidmodelindicatesthat,thenodalareaisalsoplastically deformed.Thistrendbecomesmoresignificantwithincreasing𝜑,be- causelargersolidfractionsleadtolessgeometricaldistinction𝑟^∗_𝑛∕𝑟be- tweenthenodalareaandtheligament asshownin Fig.5c.Sothat thenodalcorrectedmodeltendstopredictamuchlowerstrengthata higher𝜑.Afitofthedatausingthescalinglaw,Eq.(5)(greendashed line)leadstoaleadingconstantC_𝜎=0.72.Theincreaseoftheleading

curvatureoftheconnectedligaments.TheparameterAdefinesthefrac- tionoftheamplitudeoftheequallydistributedrandomdisplacementin relationtotheunitcellsizeoftheundistorteddiamondunitcell.The effectoftherandomizationlevelonthesolidfraction𝜑,causedbythe spatiallengtheningoftheligaments,hasbeentakenintoaccountinthe followinganalyzisassuggestedby[8,29].Concerningthemacroscopic responseoftheRVE,increasingrandomizationdecreasesthevaluesof Eand𝜎y,aswellasthedegreeoflateralextension.Formoredetailswe referto[8,29].

Forthenodalcorrectedbeammodel,therandomizationcharacter- isticsarealsointroducedtothenodalareasbecausethelengtheningof theelementsappliestoallelementsofthemodel.Thisgivesrisetothe questionifthelengtheningoftheelementscausedbytherandomiza- tionshouldbekeptasisorifitshouldbeavoidedinthenodalareasof thenodalcorrectedbeammodel.Forwhatfollows,weassumethatthe nodalelementpropertiesfollowtherelationshipsthatshowninFig.5. Thismeansthatforagivenratioof𝑙^∗_𝑛∕𝑙_𝑛anincreaseintheligament lengthl_n,causedbytherandomization,leadstoaproportionalincrease inthenodalelementlength𝑙^∗_𝑛.Itisthusconsistenttoletthenodalele- mentselongateinthesamewayastheelementsthatformtheligament.

Asthepurposeoftheintroductionofnodalcorrectedbeamelementsex- clusivelyaimsatamechanicallyequivalentbehaviorwithreferenceto thesolidmodel,thecalculationofthesolidfractionremainsunaffected.

Allsimulationsforrandomizedstructuresarecarriedoutforc_R=1.1.

EachdatapointplottedinFig.10representstheaverageoffivereal- izations; thesizeof theerrorbar correspondstothestandarddevia- tion.Fig.10ashowstheratioofthemacroscopicYoung’smodulusof thenodalcorrectedbeammodel,E_NCBM,tobeammodel,E_BM,withre- specttovariousrandomizationlevelsfromA=0thatisperfectordered toA=0.3thatisstronglydisordered.Itisshownthatforallrandom- izationlevelsthestiffnessratioincreasesnearlylinearlywiththesolid fraction,indicatingthecontributionofthenodalmasstothestiffness.

Moreover,withincreasingrandomizationlevelthecurvesshiftrightand down.Therightshiftisduetotheincreaseofsolidfractionthatiscaused bytheincreaseoftherandomizationlevel.Thetrendofslightlyshift- ingdownwardsrevealstheeffectofthedegreeofrandomizationonthe complianceinthenodalareas,whichisslightlyreducingthenodalmass contributiononthemacroscopicelasticity.Thiseffectisabout10%if weconsiderthefullrangefromzerotomaximumrandomization.

The verticalaxisof Fig. 10bdemonstrates theratio of theyield strengthofthenodalcorrectedbeammodel𝜎y,NCBMtothebeammodel 𝜎y,BM withrespecttovariousrandomizationlevels.Thegeneraltrend ofFig.10btoaisopposite,namely,increasingsolidfractionleadstoa relativedecreaseofthecontributionofnodalmasstothemacroscopic strength.Forbetterunderstandingofthis effect,thelinkagebetween thecurrentdiscussiontothediscussionsofFig.5a,bandcisrequired.

InthediscussionofFig.5itwasstatedthatwithincreasingr/lthegeo- metricalgapbetweenacylindricalshapednodeandanactualspherical noderesponseisclosingwithincreasingsolidfraction,i.e.𝑙_𝑛^∗∕𝑙_𝑛→1and 𝑟^∗_𝑛∕𝑟_𝑛→1.Itcanbealsobeobservedthatthestrengthratiogradually shiftsdownwardsfrom2.5to2forA=0andfrom1.8to1.5forA=0.3.

Becausetheplasticdeformationinitiatesinthetransitionfromthenodal

Fig. 10. Results for the macroscopic properties of the randomized RVE as ratio of the nodal corrected beam model (NCBM) to the beam model (BM) (a) Young’s modulus; (b) yield strength.

Fig. 11. Determined yield strength of the solid phase vs. ligament size, L . Experimental data are taken from [32,33] ; model data from [29] .

areatotheligament,theeffectonthestrengthreduceswithreducing sizeofthenodalelements.Withincreasingdegreeofrandomization,the correlationbetweenlocalgeometryandmacroscopicstrengthismore andmorereducedandbecomesweakforA=0.3andsolidfractionsof 𝜑≤0.3.

4.2. Analysisofexperimentaldata

Toevaluatetheperformanceofthenodalcorrectedbeammodelthe sizedependentyieldstrengthoftheligamentsisdeterminedfromlit- eraturedataonmacroscopiccompressionofnanoporousgoldsamples withdifferentligamentsize.

Intheworkof[29],thescalingrelationforthemacroscopicyield strengthwasusedasthestartingpointforfittingthetruestress-strain curvesobtainedfrommacroscopiccompressingnanoporousgoldsam- pleswithvariousligamentsize[31,32]usingtheRVEmodel.Theout- comesthatwerefromthefittingwereadatasetofyieldstrengthsforthe solidphase𝜎y,Sindependenceofligamentsize,representedbyhollow circlesinFig.11.Thisdatasetisplottedagainstresultsfromtheworks of[32,33]assolidblacksymbolsinFig.11,thatweredeterminedus- ingindependentapproaches.Roschningetal.’sresultsareofthesame magnitudeasthosedatareportedinthereferencedliteratures,butone resultforthe50nmligamentsizeappearedtobeanoutlierwhichcould notbeexplainedby[29].Theonlydifferencecomparedtotheother datawasthatithadthehighestsolidfractionofallsamplesof𝜑=0.3.

Were-analyzedthedatawiththenodalcorrectedRVEfollowingthe sameapproachproposedby[29]witharandomizationlevelofA=0.23 andaligamentaspectratioofr/l=0.25.Theresults,shownassolidred symbolsinFig.11,leadtoanelevationofRoschningetal.’sresults.The nodalcorrectedbeamRVEseemstoprovideanoverallcloseralignment withtheresultsfrom[32,33].Inparticular,thepreviousoutlierisnow closertotheoveralltrend.

Thedifferentelevationintheyieldstrengthforthedifferentdata pointscan beunderstoodwiththehelpofFig.9b.Thenodalcorrec- tionappliedin[29]wasbasedonacorrectiontermwhich– asaresult oftheassumedrigidityofthenodalmass– increasinglyoverestimates thegeometricalstrengtheningforincreasingsolidfraction.Becausethe materialsyield strengthandthegeometrical strengtheningeffect are multiplicativewithrespecttothemacroscopicstrength,areductionof thegeometricalstrengtheningeffecttotheaccuratevaluemustbecom- pensatedbyanincreasedyieldstrengthofthesolidfraction,𝜎y,S.The higherthesolidfraction,themoretheidentifiedyieldstrengthisele- vated. Consequently,theoriginaloutlier,whichhasthehighestsolid fraction,getsmostlyelevated.

5. Conclusions

Inthecurrentwork,anewbeammodelingconceptfornanoporous metalshasbeenproposedforpredictingthemacroscopicelastic-plastic responsebyincorporatingtheconnectingnodalmassinthejunctions ofthenetworkstructure.Thenodalcorrectedbeammodeliscalibrated basedonthemechanicalresponseforbendingofabeamtetrahedron structureinrelationtoitscorrespondingsolidmodelbyadjustingthree additionalparametersspecificallyassignedtothenodalarea.Anexcel- lent agreementbetween thenodalcorrectedbeamandsolidRVEfor elasticmodulus andyieldstrength hasbeenachieved, whichis con- ducted withoutlosing theadvantage of thehighcomputationaleffi- ciency.

Thenodalcorrectionleadstoanincreaseinthemacroscopicstiff- nessandstrengthofthebeammodel.Theleadingconstantintheresult- ingscalinglawforYoung’smodulusisC_E=0.57and,comparedtothe beammodelwithoutnodalcorrection,aboutafactoroftwolarger.It ishoweverstilllessstiff thantheoriginalGibson-Ashbymodelthathas aleadingconstantofC_E=1.Theincorporationofthenodalmassdoes notchangetheexponentof2inthescalinglaw.

Withrespecttomacroscopicplasticdeformation,thecorrectionsug- gestedin[8,29]significantlyoverestimatestheeffectofthenodalmass for solid fractionsabove 15%. Itwas found, thatthescaling lawof Gibson-Ashbyverywellfitsthedatafromthesimulationswiththenodal correctedRVE,whentheleadingconstantissettobeC_𝜎=0.72.Thisdif- ferenceto[8]isnowunderstoodasaconsequenceoftheoverestimation

SupportwasprovidedbyDeutscheForschungsgemeinschaftwithin SFB986‘‘Tailor-MadeMulti-ScaleMaterialsSystems:M³”,projectB4.

AppendixA.Nomenclatures

Parameters Descriptions 𝜑 Solid fraction

w Transverse loading displacement

R Nodal radius

l _n Length between nodal center and the end of the ligament attached to the node

𝑟 ^∗_𝑛 Larger cross-section radius of beam element in the nodal area (adjustable)

𝑙 ^∗_𝑛 The length between the node center and the end of the ligament (adjustable)

r Ligament radius

l Ligament length

c R Constant governing nodal mass

R Nodal radius

E S Young’s modulus 𝜈 Poisson’s ratio 𝜎_y,S Yield strength

k _solid Stiffness of the solid tetrahedron F _solid Solid tetrahedron strength k beam Stiffness of the beam tetrahedron F beam Beam tetrahedron strength

𝐸 _𝑛^∗ Young’s modulus for the beam elements of nodal area (adjustable) a 0, a 1, and a 2 Constants for fitting functions Eqs (1) –(3)

𝜀 _eng RVE macroscopic compression engineering strain C _Eand C _𝜎 Constants of Gibson-Ashby model

A Randomization parameter

E NCBM Macroscopic Young’s modulus of nodal corrected beam model E BM Macroscopic Young’s modulus of beam model

𝜎y,NCBM Macroscopic yield strength of nodal corrected beam model 𝜎y,BM Macroscopic yield strength of beam model

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