AMICROMECHANICALPLATFORMTOSTUDYNONLINEARBEHAVIOROF ELASTOMERICMATERIALS By VahidMorovati ADISSERTATION Submittedto MichiganStateUniversity inpartialoftherequirements forthedegreeof CivilEngineering-DoctorofPhilosophy MechanicalEngineering-DualMajor 2020 ABSTRACT AMICROMECHANICALPLATFORMTOSTUDYNONLINEARBEHAVIOROF ELASTOMERICMATERIALS By VahidMorovati Cross-linkedpolymersdemonstratenonlinearbehaviorunderlargedeformationswithinelastic featuressuchastheMullinseffect,permanentset,deformation-inducedanisotropy,andprogres- sivestresssoftening.Whileseveralconstitutivemodelsaredevelopedtotakeintoaccounteachof thosefeaturesindividually,thereareonlyafewmodelswhichcanconsiderdamageaccumulation incross-linkedelastomersthatoccurduetomultipleparallelfactors. Here,anewmodularplatformispresentedtointegratedifferentinelasticmechanismsintoone generalizedconstitutivemodel.Theconceptofnetworkdecompositionisthekeystoneofthepro- posedplatform.Basedonthisconcept,thepolymernetworkisconsideredasacombinationof parallelnetworks,eachresponsiblefortheinelasticresponse.Theenergyofeachnet- workiscalculatedthroughtheconceptoftheunitsphere.Consequently,thepolymermatrixtotal strainenergycanbeestimatedbysummationofthefreeenergyofthesub-networksinalldirec- tions.Therefore,athree-dimensional(3D)polymermatrixcanbedecomposedtounidirectional sub-networkelementsuniformlydistributedoveraunitmicro-sphere,whichhostsa1D inelasticmechanism.Thenetworkmodelscanbesubstituted,upgraded,orremovedwithout encingtheintegrityoftheframework.Inordertoimprovetheaccuracyoftheproposedframework, thetheoryofelastomerelasticityhasbeenrevisited.Next,differentmicro-mechanicalmodelsare developedtodescribethenatureofMullinseffect,permanentset,deformation-inducedanisotropy, andneckinginstabilityinhighlycross-linkedelastomericgelsbasedondifferentconcepts. First,thepopularassumptionsthatcomputationalaccuracyandsimplicityofthepro- posedframeworkareexamined.Inmodelingpolymericsystems,twocompetingfactorsdetermine thetypeofmaterialmodelthatshouldbeusedinthesimulation:computationalcostandaccu- racy.Optimizingthetrade-offbetweenthesetwofactorsdeterminestheminimumrequirementsof themodel.Theproposedmodularplatformenablesustoselectthenetworksbasedonthistrade- off.Furthermore,networkmodelsaredesignedtoreturnstrainenergy;thescale-transitionwillbe basedonamicro-sphereconcept,andtheNon-Gaussianentropicbehaviorisassumedforpolymer chains.Thenon-gaussiantheoryisoftenapproximatedbytheKuhn-Grün(KG)distributionfunc- tion,whichisderivedfromtheapproximationofthecomplexRayleigh'sexactFourier integraldistribution.TheKGfunctioniswidelyacceptedinpolymerphysics,wherethenon- Gaussiantheoryisoftenusedtodescribetheenergyofthechainswithvariousxibilityratios. However,theKGfunctionisshowntoberelevantonlyforlongchainsandbecomesextremely inaccurateforchainswithfewerthan40segments.Inordertoovercomethisshortcoming,anovel ofnon-GaussiantheoryusingtheinverseLangevinfunctionisdevelopedtoprovidea familyofapproximationfunctionsfornon-Gaussiantheorywithdifferentdegreesofaccuracy.In addition,asetofsimpleandaccurateapproximationoftheinverseLangevinfunctionisproposed tofurtherimprovetheaccuracyoftheenergyofa1Dpolymerchain. Next,twoconstitutivemodelsaredevelopedtounderstandanddescribethemechanicalbehav- iorofdoublenetworkhydrogel(DNgel)basedonstatisticalmicro-mechanicsofinterpenetrating polymernetworks.Inthemodel,thenonlinearbehavioroftheDNgelsisattributedtothe existenceofpre-damageinthenetworkduetoswellingduringthepolymerizationprocess. Inthesecondmodel,DNgelsbehaviorisdividedintothreepartsincludingpre-necking,necking, andhardening.Thenetworkisdominantintheresponseofthegelinthepre-neckingstage. Thebreakageofthenetworktosmallernetworkfractions(clusters)inducesthestresssoften- ingobservedinthisstage.Thedisentanglementofthesecondnetworkchainsfrombroken networkchainsandlongchainsinthesecondnetworkarealsoconsideredasmaincontributors totheresponseofgelsinneckingandhardeningstages,respectively.Thecontributionofclusters decreasesduringtheneckingasthesecondnetworkstartshardening.Thenumericalresultsofthe developedmodelsarevalidatedandcomparedbyuni-axialcyclictensileexperimentaldataofDN gels.Finally,aimplementationoftheproposedmodelispresentedtosimulatethe initiationandpropagationofneckinginstability. Copyrightby VAHIDMOROVATI 2020 "Thisdissertationisdedicatedtomywife, Fatemeh , forherunconditionalsupportandlovethroughalltheseyears, also tomyrespectfulparentsandparents-in-lawforalwaysbelievinginme." v ACKNOWLEDGEMENTS IwouldliketoexpressmydeepestgratitudetomyadvisorDr.RoozbehDargazany,forhis endlesssupport,patience,andguidanceduringmyPh.D.program.Fornotonlybeingmyadvisor, butalsoafriendwhobuiltmyvisiontowardsfuture.Hisencouragementsandhisconstantfaithin memadethisdissertationandresearchpossible.Iamalsoindebtedtomydissertationcommittee members,Dr.ThomasPence,Dr.WeiyiLu,andDr.MohsenZayernouri.Itwasaprivilege andanhonorformetohavemultiplecourseswithProfessorPenceandusehisrichexperience andexpertiseinmyofinterestasacommitteemember.IamalsothankfultoDr.Lufor hispresenceinmycommittee.Hisinsightsinhiscourseandhisvaluableinputsasacommittee membermademefamiliarwithimportantaspectsofsmartmaterialsthathelpedmeenormously duringmyPh.D.study.IamalsogratefultoDr.Zayernouriforalwaysbeingsosupportiveand friendly,andforhisinspiringguidanceandconstructivesuggestionstoimprovemydissertation. Itrulyappreciatealltheirtime,advice,insight,andguidance,whichcontributedgreatlytomy personalandprofessionaldevelopment. MysincerestgratitudealsogoestomydearpeersandfriendsintheHigh-PerformanceMa- terials(HPM)group,especiallytoHamidMohammadi,AmirBahrololoumi,YangChen,and MohammadAliSaadatforsharingideas,beingsosupportive,andhelpingmethroughoutthis process.IwouldalsoliketothankKian,Nazanin,Amirreza,AliAkhavan,MohammadYavari, MehdiSamie,Saeed,andFarnoushfortheirfriendshipandencouragement.Mysinceregratitude alsogoestomyamazingfriends,AliZockaie,Mehrnaz,Wendy,andJacobfortheirsupportsand beingmysecondfamilywhilebeingawayfrommyparentsandsiblings.Ialsohadagreattime sharinganofwithMahdiGhazavi,Mohammadreza,Aksel,Puneet,andMumtahin,whohave madeworkingenjoyableandfunforme. MyspecialthanksgotomydearwifeandbestfriendFatemehFakhrmoosavi,whohasalways beensupportivethroughoutmystudiesandhasmadecountlesstohelpmegettothis point.Withouther,thisdissertationwouldcertainlynothavebeenpossible.Last,butnotleast, vi Iamtrulygratefultomydearparents,parents-in-law,andlovelysiblingsfortheirconstantlove, support,andenthusiasticencouragement. vii TABLEOFCONTENTS LISTOFTABLES ....................................... x LISTOFFIGURES ...................................... xi CHAPTER1INTRODUCTION ............................... 1 1.1Motivation.......................................1 1.1.1Micro-mechanicalPlatform..........................3 1.2OutlineofthePresentWork..............................4 CHAPTER2MODELINGOFELASTOMERS ...................... 6 2.1SomeNotesonContinuumMechanics........................6 2.1.1Deformationgradient.............................6 2.1.2Deformationrate...............................10 2.1.3Stressmeasures................................11 2.1.4Balanceprinciples..............................13 2.2Thermo-elasticity...................................15 2.3ThermodynamicConsistency.............................17 2.4IncompressibleMaterials...............................19 CHAPTER3IMPROVEDAPPROXIMATIONSOFNON-GAUSSIANPROBABIL- ITY,FORCE,ANDENERGYOFASINGLEPOLYMERCHAIN .... 20 3.1Introduction......................................20 3.2StatisticalMechanicsTreatments...........................24 3.3ApproximationofNon-GaussianDistribution....................28 3.4EntropicForceofaSingleChain...........................32 3.5ApproximationoftheEntropicForce.........................33 3.6ApproximationoftheEntropicEnergy........................37 3.7Conclusion......................................39 APPENDIX.........................................41 CHAPTER4AGENERALIZEDAPPROACHTOGENERATEOPTIMIZEDAP- PROXIMATIONSOFINVERSELANGEVINFUNCTION ........ 45 4.1Introduction......................................45 4.2ProposedApproach..................................49 4.2.1Maclaurinapproach........................50 4.2.2Maclaurinwithoptimizedlast-term................54 4.2.3Optimizedpowerseries............................55 4.3Conclusion......................................61 CHAPTER5MICRO-MECHANICALMODELINGOFTHESTRESSSOFTENING INDOUBLE-NETWORKHYDROGELS .................. 64 5.1Introduction......................................64 viii 5.2StatisticalMechanicsofPolymers..........................66 5.2.1Non-Gaussiandistributionfunction.....................67 5.2.2Doublenetworkhydrogels..........................68 5.3ConstitutiveModel..................................70 5.3.1Networkdecomposition............................70 5.3.2Modularplatform...............................71 5.3.3Firstnetwork:Brittlenetworkwithdamage.................73 5.3.4Secondnetwork:Hyper-elasticresponse..................77 5.4Macro-scaleResponse.................................77 5.4.13Dgeneralization...............................77 5.5ThermodynamicConsistency.............................80 5.6Analysis........................................82 5.7ModelPredictionsvs.ExperimentalResults.....................85 5.8ConcludingRemarks.................................90 CHAPTER6APHYSICALLY-BASEDMODELOFSTRESSSOFTENINGWITH NECKINGPHENOMENAFORDOUBLENETWORKGELS ...... 92 6.1Introduction......................................92 6.2DoubleNetworkPolymers..............................94 6.2.1Neckinginstability..............................95 6.2.2Networkdecomposition............................97 6.3Micro-mechanicsofaSingleChain..........................101 6.4StatisticalNetworkModel..............................102 6.4.1Probabilityofchainlengthbetweencross-links...............103 6.4.2Distributionalterationduetodamage....................105 6.4.3Totalenergyofsub-network.........................108 6.5Hyper-elasticModel..................................108 6.6ConstitutiveModel..................................109 6.7FiniteElementLinearization.............................111 6.8ModelValidation...................................114 6.9NumericalSimulations................................116 6.10Conclusion......................................121 APPENDIX.........................................123 CHAPTER7SUMMARYANDFUTUREWORKS .................... 129 7.1GeneralRemarks...................................129 7.2PotentialFutureResearch..............................131 BIBLIOGRAPHY ....................................... 133 ix LISTOFTABLES Table3.1.Relativeerrorofapproximateddistributionfunctionbyconsidering1and 2termsforchainswithdifferentlength...................31 Table3.2.Max.relativeerrorofapproximateddistributionfunctionsforchainswith differentlength................................37 Table3.3.SummaryoftheproposedapproximatesforPDF,entropicforceandstrain energyalongwiththeirrelativeerrorswithrespecttoexactones......40 Table4.1.MaximumrelativeerrorofEq.4.2.4,4.2.5,4.2.6and4.2.8withdifferent numberofterms...............................52 Table4.2.versionofEq.4.2.4and4.2.5withdifferentnumberofadded termsbyusingsinglepointerrorminimization...............55 Table4.3.Approximationswiththemainfunctionof a 0 y 1 y + P n i =1 a i y i .......61 Table4.4.Approximationswiththemainfunctionof a 0 2 y 1 y 2 + P n i =1 a i y 2 i 1 .....62 Table4.5.Approximationswiththemainfunctionof a 0 ˇ 2 tan ˇ 2 y + P n i =1 a i y 2 i 1 ...63 Table5.1.Thereferencesetofparametersoftheproposedmodel...........85 Table5.2.Parametersoftheproposedmodeltotheonecycleoftensiletest performedonaDNgelin[1]........................89 Table5.3.Parametersoftheproposedmodeltothetensiletestperformedona differentsetofDNgelsdata.........................89 Table6.1.Materialparametersoftheconstitutivemodeltothetensiletestper- formedonadifferentsetofhydrogelsdata.................117 x LISTOFFIGURES Figure1.1.(a)Aschematicalstress-stretchbehavioroftherubber-likeelastomers underauni-axialtensiontest.Inelasticeffects(Mullinseffect,permanent set,andhysteresis)(b)Cyclictensiontestintransversedirectionofprevi- ouselongation.Inelasticeffects(Deformationinducedanisotropy)(c-d) Theidealizedoftheconstitutivebehaviorofelastomersby excludingpermanentsetandhysteresis...................3 Figure2.1.Deformationandmotionofacontinuumbody................7 Figure2.2.Continuummediumwithasurfaceelementandcorrespondingforcevectors.12 Figure3.1.ComparisonbetweenExactandKG(a-b)distributionfunctionsandstrain energies(c-d)forchainswith n =8 and 64 andtheirrelativeerrors....23 Figure3.2.ComparisonbetweenGaussian,KG,AmendedKGandEq.3.11distri- butionfunctionwiththeexactPDF(RelativeerrorofPDF a,cande and Relativeerrorof ln( P n ) b,dandf )ofidealchainswithdifferentlengths a-b n =8 , c-d n =16 and e-f n =64 ....................29 Figure3.3.Therelativeerrorofapproximateddistributionfunctionrespecttothe exactPDFforchainswithdifferentnumberofsegmenta) m =1 andb) m =2 .....................................31 Figure3.4.Comparisonoftheentropicforceofasinglechainresultedfromexact non-Gaussiandistributionfunctionanditsapproximations,steepestde- centapproximation,andinverseLangevinfunction(a-b),andtheirrela- tiveerrorswithrespecttotheexactfunction(c-d)foraandc) n =8 and bandd) n =64 ................................33 Figure3.5.TherelativeerroroftheILFa)withrespecttothelengthofchainandb) Averagednormalizedforalllengths.....................35 Figure3.6.Therelativeerroroftheproposedentropicforcefora) n =8 andb) n =64 alongwiththerelativeerroroffullsteepestdecentapproximation.36 Figure3.7.Themaximumrelativeerroroftheproposedentropicforceforchains withdifferentlengthalongwiththemaximumrelativeerrorofKGap- proximation.................................36 Figure3.8.Therelativeerrorsofapproximationsofentropicenergy(Eq.3.24a, 3.24band3.26)respecttotheexactentropicenergyforchainswithdif- ferentnumberofsegmentsa) n =8 ,b) n =16 ,c) n =32 andd) n =64 ....................................38 xi Figure4.1.Relativeerrorofa)Eq.4.2.4andb)Eq.4.2.5withdifferentnumberof terms.....................................52 Figure4.2.Relativeerrorofa)Eq.4.2.6andb)Eq.4.2.8withdifferentnumberof terms.....................................53 Figure4.3.One-dimensionalvariationofvalueoftheobjectivefunctionrespectto thecoefofmainfunction( a 0 )fordifferentformulainEq.4.2.13..57 Figure4.4.RelativeerroroftheILFapproximationfunctionwithrespecttotheco- ef a 0 and a 1 forWarner-likeformulainEq.4.2.14..........58 Figure4.5.Minimummaximumrelativeerrorsofdifferentorderpolynomialfor y 1 y mainfunctionincomparisonto[2]......................59 Figure4.6.MaximumrelativeerrorsofthedifferentfractionalestimationsoftheILF L 1 ( y )= n i =1 a i y i m j =1 b j y i ..............................60 Figure5.1.SchematicviewofstructuralcompositionoftheDNgelsasthesuperpo- sitionoftheandsecondnetworks.Thenetworkisnotxible andhighlycross-linked.Thesecondnetworkisaloosenetworkandis highlyxibleduetoitslongpolymerchains................69 Figure5.2.ConstitutiveresponseofaDN-hydrogelspecimenunderquasi-staticcom- pression,whichincludethreetypicalinelasticfeatures,i.e.,stress-softening, primarycurveandhardening.Nohealingbehaviorhasbeenreported.[1].70 Figure5.3.Theschematicpictureofthetwo-steppolymerizationmethodtoprepare DNhydrogelsandthedamagecausedbyswelling.............71 Figure5.4.Schematicbreakdownofamodularframeworkconsistingoftwonetwork modelsfortheandsecondnetworkstoillustratetheconceptofthe networks,andsub-networks.Thesub-networkconsistsadistribution ofshortschainwithdifferentlengthandaverageend-to-enddistance, R andthesecondsub-networkconsideredtobeasetoflongchainswith samelengthandend-to-enddistance.....................72 Figure5.5.StresssofteningintheDNgelsinthecourseofdeformation.(a)schematic viewofdifferentfragmentsofthenetworkconnectedbythechains ofthesecondnetwork,(b)internalstructureofafragmentintherefer- encestate,(c)decomposedsubstructureofafragmentinwhichshorter chainsweredebondedduetheapplieddeformation.(d)schematicrep- resentationofchainlengthdistribution,whichshowsexistenceofshort chainsas1 st networkandlongchainsas2 nd network...........75 xii Figure5.6.Theoftheparameters R;n max ; p onthemechanicalresponse ofhydro-gels.Redsolidlinesrepresentthereferencecurve,andthe dashedlinesshowthechangesinmaterialresponseduetothevariation ofthecontrolparameters..........................84 Figure5.7.Effectofthefourparameters( n ;˙ n ; and ˘ )ofthe1 st networkonthe materialresponseinthecourseofuni-axialtension.............86 Figure5.8.Effectofthe2 nd networkparametersonthelargedeformationresponse ofuni-axialtension.Redsolidlinesrepresentthereferencecurve,andthe dashedlinesshowthechangesinmaterialresponseduetothevariation ofthecontrolparameters...........................87 Figure5.9.Comparisonofthenominalstress-stretchcurvesofthemodelandthe experimentfortheuni-axialtensiletests...................88 Figure5.10.Comparisonofthenominalstress-stretchcurvesofthemodelandthe experimentfordifferentDNgels.......................90 Figure6.1.SchematicsofaDNgelnetwork,decompositionofitsnetworksandtheir chaindistributions.Thehighlycross-linkednetworkwithshortand brittlechainsandloosesecondnetworkwithlongandxiblepolymer chains....................................95 Figure6.2.SchematicviewoftheconstitutivebehaviorandchainunzippingofaDN gelunderquasi-statictensionshowinginelasticfeaturessuchasstress softeningafterloading,permanentdamageandneckinginstability..97 Figure6.3.Schematicframeworkofproposedmodelconsistingofthreesub-networks. Theandinteractionnetworksconsideredasastochasticdamageable modelsandthesecondnetworkasahyper-elasticnetwork.........100 Figure6.4.ThemaximumrelativeerroroftheKGandenhancedKGforchainswith differentlength................................102 Figure6.5.Schematicrepresentationofacross-linkednetworkandprobabilityof existenceofcross-linkattheendofeachsegment.............104 Figure6.6.Effectof p c and R 0 ontheprobabilityofpolymerchainexistencewith n segmentsbetweencross-links........................105 Figure6.7.Theprobabilitydistributionalterationinthecourseofdeformationand theschematicviewoflongerchainsactivatedduetodetachmentofthe shorterchains.................................107 Figure6.8.Comparisonofthenominalstress-stretchcurvesofthemodelandthe experimentfortheuniaxialtensileexperiment[3]..............116 xiii Figure6.9.Comparisonofthenominalstress-stretchcurvesofthemodelandthe experimentfortheuniaxialtensileexperiment[4]..............117 Figure6.10.Themodelpredictionforpre-neckingandfullresponseofthematerial alongwithcontributionofeachnetworkintotalresponseof[5]......118 Figure6.11.TheelementsimulationoftheneckingofDNhydrogels;geometry andboundaryconditions...........................119 Figure6.12.TheelementdiscretizationoftheneckingofDNhydrogels;a) Aligned2x2,b)Aligned4x4,c)Unalignedcoarse,d)Unaligned...119 Figure6.13.TheNeckingofDNhydrogels:Acomparisonofstress-stretchcurves betweentheexperimentalresultsandtheelementsimulation....120 Figure6.14.TheelementsimulationoftheneckingofDNhydrogels;effectof ,i.e.imperfectionsize,ontheresponse..................120 Figure6.15.TheelementsimulationoftheneckingofDNhydrogels;various stagesofnecking...............................121 xiv CHAPTER1 INTRODUCTION 1.1Motivation Elastomericmaterialscanbefoundinabroadrangeofproducts,rangingfrombio-medicaldevices totiresandhoses.Elastomershaveahighextensibilitywithnonlinearelasticbehavior,high-energy absorptionability,dynamicdampingcapacity,andlowcostofmanufacturing.Thesecharacteris- ticsmakethemanidealoptionfordifferentengineeringapplications.Inaddition,themechanical performanceofthesematerialscanbeenhancedbydifferentstrategieslikesynthesiz- ingmulti-networkelastomersoradditionofsuchascarbonblack,silica,andNanoparticles. Theelastomerbehaviorisnon-linearandverycomplicatedduetohighstretch-ability,nearlyin- compressibility,permanentdamage,andtime-temperaturedependentbehavior.Althoughrecent advancesintheprocessandcharacterizationoftheelastomershaveledtoantimprove- mentsintheirproperties,ourunderstandingoftheloadtransfermechanismwithinthemhasre- mainedsparseandinconclusive.Availabilityofanaccurateconstitutivemodeltopredictbehavior ofelastomersunderdifferentloadingconditionsisimportantinthedesignprocedure. Mechanicalpropertiesofrubber-likeelastomerswerestudiedintensivelyduringthelastcen- tury[6,7,8,9,10,11,12,13,14].Thebehaviorofelastomersincyclicdeformationincluding uni-axialtension,compressionandsheartestsshowsmanycomplexandinterestingfeatures.Fig. 1.1-ashowsatypicalstress-stretchbehaviorofledrubbersunderuni-axialloading.Inauni-axial tension,aconsiderablestress-softeningisobservedbetweenloadingandunloading.Theamount ofthissofteningreducesinthesubsequentcycleuntilitreachesastabilizedvalue,generallyre- 1 ferredasHysteresis.oftheconstitutivebehaviorofelastomersisaregularpractice incurrentmodelingapproaches.Inelastomers,thehysteresisisconsiderablysmallerinsecond andsubsequentdeformationcyclesandthus,itisoftenconsideredaszero[15,16].Thus,theload- ingandreloadingareassumedtobeidenticalandthehysteresisiseliminatedfromtheobtained experimentalresults(SeeFig.1.1d).Inthiscase,theunloadingcurveprovidesagoodestima- tionofthereloadingresponseofthesoftenedmaterial,andthedifferencebetweentheloadand theunloadcurvesinthecycleiscalledidealizedMullinseffect[17,18,19,20,21].Recent experimentalstudiesshowedthattensilecyclesinonedirectiondonotcausetheMullinseffect intheperpendiculardirection[22,23,24].WhentheMullinseffectintensionisalreadypresent, compressivestretchesweakenthestressinotherdirections[25].Accordingly,theanisotropyofthe Mullinseffecthighlydependsonthedeformationinotherdirections[26].Thiswork focusesondevelopingaplatformtopredicttheidealizedMullinseffect,strain-inducedanisotropy andpermanentsetbyneglectingthehysteresis.Themodularnatureoftheproposedplatform, however,allowsustolateraddnewinelasticfeaturestothepredictedresponsesuchashysteresis andstrain-inducedcrystallization. ExperimentalevidenceoftheMullinseffect,permanentset,anddeformation-inducedanisotropy suggestthatallthreeofthemcanbeconsideredaspermanentdamagemechanismsinelastomers [27,23,10].Permanentdamagehasbeeninvestigatedbyseveralconstitutivemodels[28,22,23, 29].Twomaincategoriesofproposedconceptscontainphenomenologicalandmicro-mechanically motivatedmodels.Somematerialparameterswithnophysicalinterpretationsareusedinphe- nomenologicalapproaches,whichshouldusuallybebyaprocedure.Micro-mechanically motivatedmodelsareproposedtoprovideabetterunderstandingofmechanicalbehaviorofMullins effectbytheinterpretingphysicalnatureofthephenomenon.Thesemodelsusevariousconcepts includingthebreakageofchainsbetweentherubberandtheslippingofmolecules,clus- ter'sruptureofchaindisentanglement,andmorecomplexcompositestructureformations [30,23,31]. 2 (a)uni-axialloading (b)transverseuni-axialloading (c)idealizedresponse (d)idealizedresponse Figure1.1.(a)Aschematicalstress-stretchbehavioroftherubber-likeelastomersunderauni-axial tensiontest.Inelasticeffects(Mullinseffect,permanentset,andhysteresis)(b)Cyclictensiontest intransversedirectionofpreviouselongation.Inelasticeffects(Deformationinducedanisotropy) (c-d)Theidealizedoftheconstitutivebehaviorofelastomersbyexcludingperma- nentsetandhysteresis. 1.1.1Micro-mechanicalPlatform Thisstudyproposesamodularplatformbasedonaconstitutivemodel.Themodeldescribesa polymermatrixmechanicsthroughabasisinitiatedfrommolecularphysics,andimplementsit inamulti-scalemodel.Theaccuracyimprovementoftheplatformistwo-folds,(i)enhancing thepolymerphysicstheoryusedintheplatformand(ii)increasingthenumberofmodulesto considerdifferentphenomena.Therefore,theframeworkconsistsofmodulesandseveraltheories withdifferentaccuracylevels.Eachmodulecanbeaddedfromexistingmodelstoconsidera inelasticphenomenon.Themodulescanbesubstituted,upgraded,orremovedwithout theintegrityoftheframework.Modulesarederivedtocalculatethecontributionofthe 3 sub-networkinthetotalstrainenergy.Theentropicandenthalpictotalenergycanbeconsidered forpolymerchains.Scaletransitionisperformedbasedontheassumptionofuniformspatial distributionthroughamicro-sphere3Dgeneralization.Themicro-spherescaletransition thecomplex3Dproblemtoa1Dintegrationofasimplerheologicalrepresentationofdamagefor eachspatialdirections.Micro-spherescaletransitionprovidesatechniquetomodel permanentstresssofteninginthematerial,whichcanconsiderthedeformation-induceddamage andpermanentsetasitsbyproducts. 1.2OutlineofthePresentWork Thisdissertationpresentsageneralplatformtopredictthenonlinearbehaviorofelastomericma- terialswithadesiredtradeoffbetweentheaccuracyofthemodelandcomputationalcostofits numericalimplementation.Thetrade-offisachievedthroughamicro-mechanicalplatform,which enablesustochoosedifferentphenomenatobeaddedtothemodelwithadjustableaccuracyofthe singlepolymerphysicstheory.Thecurrentstudyisdividedintosixchapters,ashortsummaryof whichispresentedinthefollowing. Chapter2 reviewsthe Non-Gaussiantheory ofpolymerphysics.Differenttheoriesarecom- paredwiththeexactdistributionofrandomwalkproblemintermsofthechainend- to-enddistributionfunction,freeenergy,andentropicforce.Finally,anewfamilyof accurateapproximationoftheNon-Gaussianprobabilitydistributionfunction(PDF), entropicforce,andthestrainenergyofasinglechainaresubsequentlydevelopedto describethemechanicsofapolymerchain. Chapter3 presentsanovelapproachwhichcanprovideafamilyofapproximationfunctionsfor inverseLangevinfunction (ILF)withdifferentdegreesofaccuracy.Thischapter startswithacurrentpracticeofestimationfortheILFandcontinueswiththegeneral propertiesofthisfunction.Attheend,threesimpleproceduresarepresented,which cantakecurrentapproximationfunctionswithanasymptoticbehaviorandenhance 4 thembytheadditionofapowerseriesoftheirinducederror.Theaccuracy/complexity trade-offforthefamilyofILFapproximations,generatedbytheproposedapproaches, iscomparedagainstthoseofotherapproachestoshowtheadvantageoftheproposed model. Chapter4 proposesa micro-mechanicalmodel tocharacterizetheconstitutivebehaviorofmulti- networkelastomersinaquasi-staticlargedeformation.Inparticular,wefocusedon describingthepermanentdamageindouble-network(DN)elastomersunderlargede- formations.Irreversiblechaindetachmentanddecompositionofthenetworkare exploredastheunderlyingreasonsforthenonlinearinelasticphenomenon.Thepro- posedmodelisabletopredictthenonlineardamagemechanismbasedonthemicro- structureofthematrix.Themodelisvalidatedwithuni-axialloadingandunloading experimentsofDNelastomers. Chapter5 presentsaconstitutivemodeltounderstandanddescribethemechanicalbehaviorand neckinginstability ofDNelastomersbasedonstatisticalmicro-mechanicsofinterpen- etratingpolymernetworks.Here,DNelastomersbehaviorisdividedintothreeparts includingpre-necking,necking,andhardening.Theconstitutivemodelforthe interaction,andthesecondnetworkisderived.Eachofthesenetworksisthemain contributorinthepre-necking,necking,andhardeningstages,respectively.Further- more,aimplementationoftheproposedmodelispresentedtosimulate theinitiationandpropagationoftheneckinginstability.Finally,thenumericalresults oftheproposedmodelarevalidatedandcomparedbyuni-axialcyclictensileexperi- mentaldataofDNelastomers. Chapter6 concludestheofthestudypresentedinthisdissertation.Possiblefutureworks extendedfromthecurrentinvestigationarealsodiscussedinthischapter. 5 CHAPTER2 MODELINGOFELASTOMERS Theconstitutivemodellingofelastomersinvolvesnonlinearcontinuummechanicalquantities, whichdescribethebehaviorofmaterialsundergoingalargedeformation.Thus,abasicfoundation oncontinuummechanics,includingfundamentalgeometricmappingsandbasicstressmeasures ofasolidbodyundergoinglargedeformations,isdiscussedinthischapter.Foracompre- hensivedescriptionandunderstandingofthetopic,thereadercanrefertocontinuummechanics referencematerialssuchas[32],[33]. 2.1SomeNotesonContinuumMechanics Incontinuummechanicstheory,itisassumedthatanobjectfullyoccupiesthespacebyitssub- stances.Physicalpropertiesofasolidormediumarerelatedthroughmathematicaltensors measures.Thesemeasuresareindependentoftheirobservedcoordinatesystemingeneral.How- ever,theycanberepresentedindifferentcoordinatesystems.Inthissection,therelationofvarious mechanicalmeasurementsarereviewed. 2.1.1Deformationgradient Acontinuumbody B t 0 ina3DEuclideanspaceattime t 0 isshowninFig.2.1,inwhichany arbitrarypoint P 0 withrespecttoanarbitrarybasiscanberepresentedby X 2 E 3 .Asthebody deformsinthespace,themediumoccupiesitscurrentat B t .Giventhatthegeo- metricalmappingofbodyfrom B t 0 to B t isonetoone,anypointfrominitial P 0 , 6 uniquelymapstoitsnewcoordinate, P ,withanarbitrarybasisof x 2 E 3 . Thegeometricalmappingoftheregionsofbodyfromcurrenton B t 0 tothereference B t inanyarbitrarycoordinatesystem, e i ( i =1 ; 2 ; 3) ,canrepresentthepositionof point P and P 0 by x = ^ x ( 1 ; 2 ; 3 ;t ) ; X = ^ X ( 1 ; 2 ; 3 )= ^ x ( 1 ; 2 ; 3 ;t 0 ) :i =1 ; 2 ; 3 ; (2.1) Therefore,thedisplacementvector, u ,ofpoint P canbewrittenas u = ^ u ( 1 ; 2 ; 3 ;t )= x X (2.2) Figure2.1.Deformationandmotionofacontinuumbody. GivenanEuclideanspacewithasetoforthonormal(Cartesian)basisvectors,say e i ( i = 1 ; 2 ; 3) ,onecanexpresseachpointofthebodyas X = X i e i ;X j = X e j ;j =1 ; 2 ; 3 ; u = u i e i ;u j = u e j ;j =1 ; 2 ; 3 ; (2.3) x = x i e i ;x j = x e j = X j + u j ;j =1 ; 2 ; 3 ; wheretheEinsteinnotation,summationoverrepeatedindices,isapplied.Deformationcanbe 7 relatedtothetangentvectorsofthecoordinatelinesineachoftheTangentvectors ofsufdifferentiable X and x canbewrittenas G i = @ X @ i = ; g i = @ x @ i ;i =1 ; 2 ; 3 ; (2.4) Therelativemotionofanarbitrarypointwithrespecttoitsadjacentpointcanbecalculatedina direction a throughthedirectionalderivativeas d ds ^ x X + s a ;t s =0 =lim s ! 0 ^ x X + s a ;t ^ x X ;t s = Grad x a : (2.5) Therefore,secondorderdeformationgradienttensor, Grad x ,canbewrittenbyalinearmapping ofvector a intovector d ds ^ x X + s a ;t s =0 as F =Grad x = d ^ x d X : (2.6) where, d X isaelementbeforedeformationand d ^ x isthesameelementonthebody afterdeformation.Thistensorplayapivotalroleindeformationkinematics,whichcandescribe therelativemotionofmaterialelementsduringdeformation.Therefore,onehastherelationof elementinreferenceandcurrentas d x = F d X ;d X = F 1 d x : (2.7) Moreover,thechangeofvolumeandsurfaceelementscanberelatedtodeformationgradienttensor thoroughtheserelations.Tothisend,avolumeinthereferenceofthreenon-co- planarvector, d X 1 , d X 2 and d X 3 ,canbeas dV 0 =[ d X 1 d X 2 d X 3 ]= d X 1 d X 2 d X 3 : (2.8) 8 UsingEq.2.7,eachofthesevectorsdeformsinthecurrentto d x 1 = F d X 1 ;d x 2 = F d X 2 ;d x 3 = F d X 3 : (2.9) Thus,thevolumeoftheelementinthecurrentcanbewrittenas dV = d x 1 d x 2 d x 3 = d x 1 d x 2 d x 3 = JdV 0 ; where J = dV dV 0 = F i :j =det F > 0 : (2.10) Inaddition,bythesurfaceelementofreferenceandcurrentstateas d A 0 = d X 1 d X 2 and d A = d x 1 d x 2 ,therelationofthesurfaceareascanbecalculatedbysubstitutingEq.2.9 into dV 0 = JdV as d A = J F T d A 0 : (2.11) where dA = j d A j and dA 0 = j d A 0 j arethesurfaceareasinthecurrentandreference respectively.Anelementlengthinreferenceandcurrentstatescanbecalculatedsimilarlyas k d x k 2 = d x d x = d X F T F d X = d X C d X ; k d X k 2 = d X d X = d x F T F 1 d x = d x b 1 d x ; (2.12) where C = F T F and b = FF T aretherightandleftCauchy-Greentensors,respectively.The changeinthelengthofalinearelement,thestretchofamaterialelement,astheratioof thedeformedtothereferencedlengthofthematerialelement.Deformationchangesthelengthof anelement d X indirection N ininitialstateto d x indirection n inthecurrentstateas 9 N = dx dX = s k d x k 2 k d X k 2 = r dX N C N dX dX 2 = N C N 1 2 ; (2.13) and n = n b 1 n 1 2 : (2.14) Anothermeasureofthechangeinelementlengthduringdeformationcanbewrittenas k d x k 2 k d X k 2 =2 d X E d X =2 d x e d x (2.15) where E = 1 2 C I = 1 2 F T F I calledtheGreen-Lagrangestraintensorand e = 1 2 I b 1 = 1 2 I F T F 1 isAlmansistraintensor. 2.1.2Deformationrate Thematerialvelocitygradientcanbesimilartothedeformationgradientby L =Grad _ x = @ @ X " @ x ( X ;t ) @t # = @ @t @ x @ X ! = _ F : (2.16) Thespatialvelocitygradient,derivativeofaspatialvelocity v withrespecttothecurrent canbewrittenas l =grad _ x = @ v @ X @ X @ x = _ FF 1 : (2.17) 10 Thespatialgradientofvelocitycanbedecomposedtoasymmetricpart d = 1 2 l + l T anda skew-symmetricpart w = w T = 1 2 ( l l T ) .Thesymmetricpartiscalleddeformationrate d = 1 2 l + l T = 1 2 _ FF 1 + F T _ F T = 1 2 F T _ CF 1 : (2.18) Inaddition,theskew+symmetricpartisreferredtospin(vorticity)tensor w = w T = 1 2 ( l l T )= 1 2 _ FF 1 F T _ F T : (2.19) TherateofGreen-Lagrangestraintensor _ E canbewrittenas _ E = 1 2 _ C = 1 2 F T dF : (2.20) Therateofthevolumechangecanbecalculatedbythetimederivativeofthedeterminantof thedeformationgradientas _ J = @ det F @t = J tr d (2.21) 2.1.3Stressmeasures Thestressisbytractionforcevector dF s perunitofarea(seeFig.2.2),whichis astheneighbouringcontinuumpointsofthebody dA .Thestressineacharbitrarypoint P 0 and P inreferenceanditscounterpartinthecurrentisbasedonareas dA and da aroundthemwiththeunitvectorsof N and n normaltothem,respectively.Thus,the relationofthetractionforceandstresscanbewrittenas d F s = T dA = t da; (2.22) wherethevectors T and t arethetractionforceinreferenceanddeformedrespec- tively.Therefore,second-ordertensors ˙ ˙ ˙ and P P P canbeexpressedbasedontheCauchy'sstress 11 theoremas Figure2.2.Continuummediumwithasurfaceelementandcorrespondingforcevectors. t = ˙ ˙ ˙ nT = P P P N ; (2.23) where ˙ ˙ ˙ representstheCauchystresstensorrelatedtoforceonthebodysurfaceinthecurrent and P P P isthePiola-Kirchhoffstresssecond-ordertensorbasedonthe surfaceareaininitialThePiola-KirchhoffstressandCauchystresstensorcan berelatedthroughtheNanson'sformula2.11and2.23as d F = t dA = ˙ ˙ ˙d A = J˙ ˙ ˙ F T d a : (2.24) ThesecondPiola-Kirchhoffstresstensors S canbecalculatedbypulling-backtheforcevector df s inthecurrentstatetothereferencetion.Similarly,anotherspatialstressmeasure,the Kirchhoffstresstensor ˝ ,canbecalculatedandconvertedtoothermeasuresasthefollowingset ofequations P = J˙ ˙ ˙ F T = ˝ F T = FS ; S = J F 1 ˙ ˙ ˙ F T ;˝ ˝ ˝ = J˙ ˙ ˙ (2.25) NotethatPiola-Kirchhoffstressisnotasymmetrictensorduetoitstwo-pointcoor- dinatesystems. 12 2.1.4Balanceprinciples ForabodyBwithvolume V ,mass M ,andboundarysurface A inthecurrentstate,therateof changeofthelinearmomentumisdirectlyproportionaltotheforceappliedonthebody,which canbeby Z M v dM = Z V ˆ _ x dV; where ˆ isthedensityand v = _ x isthevelocityvectorofaparticle.Themomentumcanbe calculatedbysummationofabodyforceandasurfaceforceas d dt Z V ˆ _ x dV = Z V f dV + Z A t dA: (2.26) ThesurfaceforcescanbeobtainedthroughtheCauchytheorem(2.23)anddivergencetheorem Z A t dA = Z A ˙ ˙ ˙ n dA = Z V div ˙ ˙ ˙dV: (2.27) BysubstitutingEq.2.27intoEq.2.26,thebalanceequationisthen Z V div ˙ ˙ ˙ + f ˆ x dV =0 : (2.28) Foranyarbitrarypointofthebody,onecanrewriteEq.2.28as div ˙ ˙ ˙ + f = ˆ x : (2.29) Inordertocalculatethebalanceofmechanicalenergy,onecanmultiplyEq.2.29withthevelocity vector v as v div ˙ ˙ ˙ + v f = ˆ v x : (2.30) 13 Eq.2.30canbefurtherconsideringsymmetryoftheCauchystresstensoras v div ˙ ˙ ˙ =div( v ˙ ˙ ˙ ) ˙ ˙ ˙ :grad v =div( v ˙ ˙ ˙ ) ˙ ˙ ˙ : d ; (2.31) whichgives div( v ˙ ˙ ˙ ) ˙ ˙ ˙ : d + v f = ˆ d dt 1 2 v v : (2.32) IntegratingEq.2.32overthevolumeofthebodyandconsideringEq.2.27yieldto d dt Z M 1 2 v v dM + Z V ( ˙ ˙ ˙ : d ) dV = Z A ( v t ) dA + Z V ( v f ) dV: (2.33) Thebalanceofmechanicalenergycanberewrittenbytakingintoaccountthephysicalmeaningof eachtermintheEq.2.33as _ K + W = P ; (2.34) where P , _ K ,and W refertothepowerofexternalforces,thestresspower,andthechangesinthe kineticenergy,respectively.Thepowerofexternalforcescanbewrittenas P = Z A ( v t ) dA + Z V ( v f ) dV: (2.35) Thekineticenergyandthestresspowerofthesystemisthenformulatedby K = Z V ˆ 1 2 v v dV; W = Z V ( ˙ ˙ ˙ : d ) dV: (2.36) Notethatthestresspowercanbecalculatedthroughotherstrain-stressmeasuresandeachofthe stressmeasures ˙ ˙ ˙ , P ,and S areconjugatewithdeformationmeasures d , _ F ,and _ E ,respectively. Forinstance,thestresspowercanbecalculatedusingthePiola-Kirchhoffstressanddeforma- 14 tiongradientas W = Z V 0 P : _ F dV 0 = Z V J 1 P : _ F dV: (2.37) Thus,thestresspowercanalsobeobtainedfrom P : _ F =( ˙ ˙ ˙ : d )= S : _ E = S : 1 2 _ C : (2.38) Inaddition,thetotalenergydissipationduetothedeformation D canbeobtained D = W Z V 0 _ dV 0 (2.39) where _ referstotherateofstoredenergyperunitvolumeinthereferencestate. 2.2Thermo-elasticity Therelationsbetweenmechanicalandthermalenergyiscalledthermo-elasticity,whichcanbe derivedusingtheandsecondlawsofthermodynamics.The1 st lawofthermodynamicsisthe conservationofenergy,whichisformulatedas dU = dQ + dW: (2.40) where dU , dQ ,and dW arethechangesintheinternalenergyofthesystem,theheatabsorbedby thesystem,andtheworkdoneonthesystem.The U and W canbeexpressedas U = Z V ˆ U dV; W = dW dt (2.41) where U istheinternalenergydensity.Accordingtothesecondlaw,thechangesintheentropyof areversibleprocess(elasticdeformation), dS atabsolutetemperature T isformulatedas dS = dQ T : (2.42) 15 Moreover,thechangeintheHelmholtzenergy A isas d A = dU SdT TdS: (2.43) Ataconstanttemperature,thechangeinHelmholtzenergyisequaltothechangeintheworkdone onthesystem dW = d A .Inthesmalldisplacement dl ofsolid-likestructures, dW duetoforce f canbewrittenby dW = fdl p h dV; (2.44) where p h and dV denotethehydro-staticpressureandthevolumechange.Inthecaseofincom- pressiblemateriallikeelastomers, dV canbeneglected.So,theforceofanisothermalprocesscan beobtainedusingEq.2.43andEq.2.44as f = dU dl T dS dl : (2.45) ThepartofEq.2.45representstheenergeticinteractionsofasinglemoleculeorthevolume changeofthebody[11].However,experimentalevidencessuggestthatthecontributionofener- geticforceinthetotalforceisnegligibleinmoderateandlargedeformationsofelastomers.Thus, theforceofthesystemcanbeapproximatedonlybyentropycomponentas f = T dS dl : (2.46) Foranentropicmaterial,theHelmholtzfreeenergyrequiredtoperturbtheentropyofthesystem is d A = TdS: (2.47) 16 2.3ThermodynamicConsistency Accordingtothesecondlawofthermodynamicthatisusedincontinuummechanics,elasticenergy ofthesystemmainlyincreasesduetothedecreaseinentropy,whichisknownasClausius-Duhem inequality D 0 : (2.48) TheconstitutivemodelisthermodynamicallyconsistentifandonlyifClausius-Duheminequal- ityisonallpointsofbodyatanytime.Inotherwords,theenergybalanceshouldbe duringathermodynamicaldeformationprocess.Theenergybalancecanberewrittenby substitutingEq.2.40toEq.2.34as _ K P = d dt ( U Q )= _ U _ Q; (2.49) wherethethermalpower _ Q isas _ Q = Z V ˆ Q dV Z A ˆ q n dA: (2.50) InEq.2.50 ;Q and q arethetransferredheatandthesurfaceheatperunitofmass,respectively. Moreover,Eq.2.49canberewrittenas ˙ ˙ ˙ : l = ˆ _ U ˆ Q + div q : (2.51) Therefore,Clausius-Duheminequalitycanbeobtainedby ˆ D = ˆT _ S ( ˆ Q div q ) 0 ; (2.52) 17 where S istheentropydensity.Clausius-Duheminequalitycanbefurtherbysubstitution oftheenergybalancelaw(2.51)andmassconservation( ˆ 0 = Jˆ )as D = T _ S _ U 1 ˆ 0 P P P : _ F 0 : (2.53) consideringthestoredenergydensityperunitofreferencevolume = ^ ( F ;T; asafunction ofsetofinternalvariables .Inmostoftheconstitutivemodels,theinternalvariableshavebeen adoptedtodescribethehistorydependentdissipativeeffects.Notethatallthequantitiesofthe Clausius-Duheminequality,suchas S; q ,and P shouldbefunctionsofinternalvariables.Fora morecomprehensivereviewofthermo-elasticity,thereaderisreferredto[34,35].Substitutingthe Helmholtzfreeenergy = ˆ 0 ( U T S ) intoEq.2.53resultsin ˆJ D = _ ˆJ S _ T + P P P : _ F 0 (2.54) where _ ( F ;T; canbeobtainedas _ = @ @ F : _ F + @ @T _ T + @ @ _ : (2.55) Eq.2.54canberewrittenconsideringEq.2.55as ˆJ D = @ @T _ T @ @ _ ˆJ S _ T + P P P @ @ F : _ F 0 : (2.56) Eq.2.56canbefurtherforthecaseofisothermalprocessandconsideringaphysical expressionforthePiola-Kirchhoffstress P P P = @ @ F ,whichleadstoaninequalityofinternal energydissipation D as @ @ _ 0 : (2.57) Thisinequalitymustalwaysbeoverthebodyduringdeformation. 18 2.4IncompressibleMaterials Resistanceofmostofelastomerstochangevolumeknownasincompressibilityconditionshould beconsideredintheconstitutivemodelling.Therefore,theinternalenergyfunctionofanincom- pressiblematerialisby = ^ ( F ;T; p ( J 1) ; (2.58) where J =det F = dV dV 0 =1 istheincompressibilityconstraintand p isaLagrangemultiplierto satisfytheboundaryconditions.Therateofthestrainenergyfunction is _ = @ ( F ;T; @ F p F T : _ F ; (2.59) wheretheincompressibilityconditionisconsideredasthevolumeremainsconstant, _ V = _ J =0 . SubstitutingEq.2.59toEq.2.39,onehas Z V 0 P dV 0 Z V 0 @ F ^ ( F ;T; p F T dV 0 : _ F 0 : (2.60) Eq.2.61shouldbeheldforeverydeformationrate, _ F ,whichleadsto P = @ ^ ( F ;T; @ F p F T ; (2.61) Thisequationisvalidforthecaseof Ł Incompressiblehyperelasticmaterial Ł Negligiblecontributionofinternalenergyindeformation Ł Isothermaldeformation Ł Moderateandlargerangeofdeformation 19 CHAPTER3 IMPROVEDAPPROXIMATIONSOFNON-GAUSSIANPROBABILITY,FORCE,AND ENERGYOFASINGLEPOLYMERCHAIN 3.1Introduction Incomputationalsimulationsofpolymericsystems,twocompetingfactorsdeterminethetypeof thematerialmodelthatshouldbeusedinthesimulation;computationalcost(i.e.thesimulation time)andtheaccuracy.Optimizingthetrade-offbetweenthesetwofactorsdeterminesthemini- mumrequirementsofthemodel.Inmechanicsofpolymers,theexcessivecomputationalcostsof accuratemodelspreventsthemfrombeingusedinlarge-scalesimulations.Here,ourgoalistopro- poseafamilyofeffectiveapproximationfunctionswithdifferentrangeofaccuracyandcomplexity thatcanaddresstheexistingtrade-offproblem. Inpolymerphysics,micro-mechanicalconstitutivemodelsaremostlyderivedfromthenon- Gaussianstatisticaldistributionofarandomlyjointedmolecularchain[36,37,38,39].Inthese models,theelasticityofthechainsisinducedfromthechangesintheprobabilityofchainend-to- enddistance, r ,inthecourseofdeformation,andthusthechangeofthechainentropy[11].The PDFofaperfectlyxiblechainwithedendpositions P ( r ) canbecalculatedusingasolution thatisproposedtosolvetherandomproblem[40,41,42].Theconceptwaslaterused inseveraltheoreticalandexperimentalstudiestodescribethepropertiesofdilutepolymersolu- tions.Indilutesolutions,theisolationofpolymermoleculesallowscharacterizationofindividual molecules.Astrongcorrelationwasfoundbetweenthenumberofsegmentsofachain, n ,andits end-to-enddistance, r givenby r / p n [11].Thiscorrelationlaterbecamethebasistoconsider 20 P ( r ) similartothatofaproblem. Inmoststatisticalpolymermodels,thestress-strainrelationforthepolymermatrixoriginates frommoleculardescriptionofdeformationofsinglechains.Theattemptinunderstanding behaviourofpolymerswasbasedonastatisticalapproachtoderiveentropicconformationofa polymerchain,whichwasindependentlyproposedbyKuhn[40]andGuthandMark[43].Both theoriessuccessfullyderivedtheGaussianPDFestimationofapolymerchainfromitsentropy [44].However,itcanbeshownthatGaussianstatisticsareexactonlyforthepolymerchainswith lengthorverysmalldeformation.Later,KuhnandGrünproposedtheinverseLangevin approximationfor 'freelyjointedchain' (FJC)toaddresstheeffectofchainlengthinthe network,andreachedtopioneermodelforsinglechainstatisticsinlargedeformation.Thepopular Kuhn-Grün(KG)modeldescribesthestatisticalprobabilityofexistenceofanunconstrainedsingle chainwithanentirelyrandomorientationinspace[45].Besidesimplicity,therelevanceofthe assumptionhasmotivatedthemajorityofmodelseversincetousetheKGfunctiontoestimatenon- GaussianPDF[38,46,47,48].ThisestimationistheorderapproximationoftheRayleigh's exactFourierintegraldistributionfunction[49],andcandescribetheextensibilityofthe polymerchainsevenatlargestrains.Thismodeliswidelyacceptedintheofrubberelasticity duetoitsaccuracytocapturestheultimatestrainofpolymernetwork[50,37,51].Inpolymer physics,mostconstitutivemodelsofthepolymermatrixsuchas3-chain[52],4-chain[53],8- chain[37],thefull-networkmodels[11,54],andthemicro-macrounitspheremodels[55]are basedontheKGapproximationsofnon-Gaussiantheory,whichgenerallyincludestheinverse Langevinfunction[11].However,differentstudiesintheliteratureexamineditsrelativelylarge errorfortheshortchains[11,39]. Asitshownintheliterature,Gaussiandistribution,energyandentropicforcehaveverygood agreementwiththeoryofrubberelasticityandexperimentalevidencesinsmalldeformations.As deformationincreasesthistheorycannotpredictthebehaviorofelastomericmaterialwithlimited extensibilityaccordingly.Inordertoovercomethisshortcoming,thenon-Gaussiantheoryofrub- berelasticityproposedtoimprovethetheoryfordeformationsnearthefailureofthematerial.The 21 complexitysomeofthesetheoriesleadstoextensiveapplicationofKGtheoryindifferentareaof polymerphysicsasasimplealternativeofnon-Gaussiantheory.DespitethefactthatKGtheoryis onlyvalidforlongchains,thereareseveralstudiesthatusedthistheorywithlessthan20segments chains[55,51,29].ArrudaandBoyce(1993)[37]proposedtheiroutstanding8-chainmodelusing KGtheory,whichisvalidatedagainstexperimentaldataassumingthechainslengthaslowas8. TheFullnetworkmodel[54]thatconsidersthespatialuniformdistributionusedtheKGmodelby chainswithonly2.8segments(therelativeerrorofentropicforceforchainwith3segmentswill beshowninsection3.4,whichisreachtomorethan 100% ,Fig.3.7). Despiteitswideacceptance,KGestimationisonlyvalidforsuflargechains ( n ˛ 40) [56,44,50].WhileKGshowsgoodgraphicalagreementwithRayleigh'sexactdistributionin lowextensibility[56],ityieldserrorsinthelargeextensibility(seeFig.3.1a).Moreover, forlongchains,KGapproximationbecomesstronglyinaccurateas r nl ! 1 astheprobability approachestozero.Figure3.1-ashowsthattherelativeerrorsofashortchainandalongchain almostisthesameunlikethemoststatedintheliterature[56].Inpolymerphysics,thestrainenergy ofachain W ,whichiscorrelatedwith W / ln( P ( r;n )) ),isusedmoreoftenthanthe P ( r ) ,and thusisthesubjectofinterest.InFig.3.1-d,wehaveplottedtherelativeerrorsinapproximating W byusingKGPDF.AsitcanbeinthisthemaximumrelativeerrorofKGenergyfunction isabout25%forthechainwith8segments.However,themaximumrelativeerrorofKGenergy functioncanreachtoasmuchas100%forthechainwith3segments.Itisevidentthattherelative errorforshortchainsaremuchhigherthanthatoflongchains.Toaddressthisproblem,Jerningan andFlory[56]introducedanewapproximation,referredtoas'amendedKuhn-Grün'(A-KG),by addinganextramultiplicativetermtotheKGfunction.Duetoitscomplexity,A-KGmodelwere usedinveryfewstudiessuchastheworkofElias-ZunigaandBeatty[46]. Currently,almostallstatisticalmodelsofchainelasticityarebasedontheKGPDF.Accord- ingly,theentropicforceresultedfromKGarethefunctionofoneparameteronly;namelythe extensibilityratio, t = r L where r istheend-to-enddistanceofachain, L = nl thecontourlength. However,studiessuggestthattheentropicforceresultedfromPDFsarebytwoparam- 22 (a)NormalizedPDF (b)RelativeerrorofPDF (c)Normalizedstrainenergies (d)Relativeerrorofstrainenergies Figure3.1.ComparisonbetweenExactandKG(a-b)distributionfunctionsandstrainenergies (c-d)forchainswith n =8 and 64 andtheirrelativeerrors. eters,namely t and n .Todate,mostconstitutivemodelssufferfromthelargeerrorsinducedby theKGfunctioninpredictingthePDF,forceorenergyinthecaseofshortchains.Sofar,thereis nootherfeasibleapproximationofPDFthatcanalsocapturethebehavioroftheshortchains. Here,wedevelopedanapproachtoderiveafamilyofapproximationsforthePDF,forceand strainenergyofpolymerchains.Suchanapproximationmodelisparticularlyrelevantinconstitu- tivemodelsofpolymerchainsthatusetheILF L 1 ( r nl ) todescribetheentropicforceofachain. SincetheILFcannotbederivedexplicitly,approximationfunctionswithdifferentdegreeoferrors areusedtorepresentit.Recently,duetoimprovementofourcomputationalpower forsimulatingtheentropicenergyofthewholenetwork,accurateapproximationoftheILFhas becomeasubjectofinterest.Inthelastdecade,severalhighaccuracyapproximationswitherrors aslowas 10 4 % havebeenintroduced[57,2,58,59].WhileaccurateapproximationsoftheILF canreducetheerrorofKGenergyandforceapproximations,therestillexistsaserror 23 inthoseapproximationsduetotheintrinsicerrorassociatedwithKGPDF.Sucherrornecessitates thefutureeffortstobedirectedtowardderivinganewapproximationforPDFfunctionbefore derivingforceandenergy. Thefundamentalsofnon-Gaussianstatisticalmechanicsofpolymersarereviewedinsec- tion3.2.WeproposenewapproximationfunctionsforPDFinsection3.3.Insection3.4,theerror ofthecurrentapproximationsofentropicenergyandforceofapolymerchainiscalculatedtoshow therelevanceofnewapproximationfunctions.Finally,insection3.5and3.6newapproximation functionsforentropicforceandenergyofasinglepolymerchainisprovided.Thefunctionsshow negligibleerrorevenforshortchainsandarerelevantforalongrangeofextensibilityratios. 3.2StatisticalMechanicsTreatments Inthissection,thenon-GaussianPDFofexistenceofachain, P ( ! r ) ,withend-to-endvector ! r andcontourlength nl isreviewed.TheprobabilitydistributionofanFJCisthesameas 3-Drandomproblem,whichdescribestheprobabilityofachainendingatacertainpointat distant r .In1905,Pearsondiscussedthedistributionofpositionofamosquitoinaforest[60]. Toaddressthisproblem,severaldistributionfunctionshavebeendevelopedever-sincebasedon theFourierintegrationoftheproblem,ofwhichwasdevelopedbyRayleighin 1919byusingthediscontinuousintegralofDirichlet[49].Theprobabilityofexistenceofachain canbederivedbytakingtheFouriertransformofcharacteristicfunctionas P exact ( r )= 1 2 ˇ 2 r Z 1 0 ˆsin ( ˆr ) sin ( ˆl ) ˆl n dˆ: (3.1) Thisequationwouldbediftosolveanalyticallyforchainswithlargenumberofsegments, n> 10 .Theexactnon-Gaussiandistributionfunctionfor3,4,and6stepswerederivedby Rayleighassetsofdiscontinuouspolynomials[49].TheexactsolutionofFourierintegralof Eq.3.1,oftenreferredtoas fiRayleighexactdistributionfunctionfl ,waslaterderivedbyTreloar 24 [42]basedonthetheoryofrandomsamplingas P exact ( r )= 1 2 n +1 ( n 2)! ˇl 2 r k n r l 2 X k =0 ( 1) k 0 B @ n k 1 C A n 2 k r l n 2 : (3.2) Similarly,WangandGuth[52],Nagai[61],andHsiungetal.[62]derivedsimilarformula- tionswithdifferentmathematicalapproaches.Toavoidthehighcomputationalcostoftheexact solution(duetoitspiece-wisenature),severalapproximationmethodsweredevelopedfornon- Gaussiandistribution.Thedegreeofmathematicaldifoftheseapproximationsdependson therequiredaccuracyandthecoveredextensibilityratio( t = r nl ).TheGaussiandistribution,for example,issimpleandhasagoodagreementwiththeexactdistributionatsmall t .Itcanbe shownthattheorderapproximationof1DrandomwalkproblemyieldstotheGaussiandis- tribution(seeAppendix3.7).However,Gaussiandistributionbecomesexponentiallyinaccurate forthechainsintheirfullyextendedlength( t ˘ = 1 ).Thus,amoreelaboratedistributionfunc- tionisrequiredtocapturethenon-GaussianPDF.Ingeneral,theapproximationfunctionsthatare developedtoapproximatetheRayleighexactPDF,Eq.3.2,canbecategorizedintothreetypes(i) Taylorexpansionapproximationswhicharevalidforlongchainswithlowextensibility,(ii)Sta- tisticalapproximationwhicharevalidforlongchainsatallextensibilityratiosand(iii)Steepest decentapproximationswhicharevalidforallchainsandextensibilityratios,althoughithasahigh computationalcost[52]. (i) Taylorexpansionapproximation(TE) hasanacceptableaccuracyforlongchainsatsmall t .InthiscaseEq.3.1canberewrittenas P exact ( r )= 1 2 ˇ 2 r Z 1 0 ksin ( kr ) e ˚ ( k ) dk; (3.3) where ˚ ( k )= n ln sin( ka ) ka .Tofurthersimplifytheaboveequation, ˚ ( k ) canbesubstituted byitsTaylorexpansion ˚ ( k )= n 1 k =1 B 2 k 1 (2 s ) 2 k (2 k )!2 k ,where B n isBernoullinumber.Byusing thetermof ˚ ( k ) Taylorseries,Eq.3.3yieldsthestandardGaussiandistributionfunction, 25 P G ,as[44], A n ( ˆ ) ˘ = exp n ( ˆl ) 2 6 ! ! P G ( r )= A 0 exp r 2 2 n ; (3.4) where A 0 = 3 2 ˇa 2 n 3 2 .Byusingfullexpansionof ˚ ( k ) amoreaccurateapproximationof P exact canbeobtainedas, P T ( r )= A 0 1 3 20 n 5 10 r 2 n +3 r 4 n 2 + ::: exp 3 r 2 2 n : (3.5) Inordertofurtherenhancetheaccuracyoftheapproximation,theTaylorexpansioncanbe writtenarounditssaddlepoint[63]as, P T SP ( r ) ˘ = A 0 exp n 3 2 t 2 (1 1 n + 2 5 n 2 )+ 9 20 t 4 (1 11 5 n )+ 99 350 t 6 : (3.6) TosimplifyEq.3.6,onecanassume 1 n ! 0 forlongchains( n ˛ 40 ),andthusreduceEq. 3.6to P T SP ( r ) ˘ = A 0 exp ( n ( t )) ; (3.7) where ( t ) isafunctionofextensibilityratioonly.SinceEq.3.7isequaltothedistribution functionresultedfromTaylorexpansionofILF,onecanconcludethattheILFapproximations arealsomainlyrelevantforlongchains. (ii) Statisticalapproximation(SA) of P exact ,alsoknownasKuhn-Grün(KG)PDF,isparticu- larlyaccurateforthelargechainsinthehighlystretchedstate[11].KGPDFisintroducedin 1942throughthemaximumtermmethodofstatisticalmechanicsas P KG ( R )= c ˆ sinh( ) exp( ) ˙ n ; (3.8) where c isnormalizationfactorthatcanbe A 0 or P exact n (10 2 ) .TheILFparameter = 26 L 1 ( t ) canbeimplicitlycalculatedthroughLangevinfunctionequation, t = L ( ) coth( ) 1 .NotethatJamesandGuth(1943)[36]andFlory(1953)[64],independently derivedthesameformulationwithdifferentapproaches.InanothereffortJerniganandFlory [56]derivedanamendedversionofKGdistributionfunctionas P A KG ( r )= A 0 sinh( ) exp( ) n t = t P KG : (3.9) (iii) Steepestdecentapproximation(SD) isderivedbyWangandGuth[52]basedonthesaddle pointapproximationofEq.3.2,whichgives P WG ( r )= A 0 sinh( ) exp( ) n t 1 t 2 2 t 1 2 1+ q ( t ) n + ::: ; (3.10) where q ( t ) isafunction[52].Usingsteepestdecentapproach,anotherapproximation functionisderivedbyYamakawa[44]whichcanberewrittensimilarto P WG as 1 P SD ( r )= A 0 sinh( ) exp( ) n t 1 t 2 2 t 1 2 (3.11) Interestingly, P SD ,Eq.3.11,isthefourtermsofWangandGuthapproximationEq.3.10 and P A KG ,Eq.3.9,isthethreeterms.AsmentionedbyJerninganandFlory[56],the term h sinh( ) exp( ) i n becomesmoreandthentheothertermscanbeneglectedfor longerchainssameasKGmodel.However,theothertermshasmorecontributioninthe accuracyofthemodelforshorterchains. Todate,thereexistsnocomprehensivestudytocharacterizetheerrorinducedbyeachofthe aforementionedapproximationmethodsinpredictingPDFofchainswithdifferentlengthsexten- 1 Originalformulationpresentedin[44]is P Y ( r )=3 3 2 A 0 2 t h 1 ( sinh( ) ) 2 i 1 2 n sinh( ) exp( ) o n ,which canberewrittenas P SD byconsidering 1 t 2 2 t = 1 2 1 sinh 2 ( ) . 27 sibilityratios.Here,acomprehensivecomparisonoftheaforementionedapproximationfunctions inpredictingPDFandstrainenergyofchainswithdifferentlengthsatdifferentextensibilityratios ispresented(seeFig.3.2). P G and P T havealmostsimilarandrelativelysmallrelativewithrespecttotheexactdistribution inonlyverysmall t .Interestingly, P T SP withonlythreetermshasextremelylowrelativeerror withrespecttoexactdistributioninsmallandmoderateextensibilityratios.Notethatincreasing thenumberoftermsintheexpansionofthisapproximationcanimprovetherelativeerrorfor larger t .Asexpected, P KG ( R ) hasanegligibleerrorforsmall t whichexponentiallygrowsas t tendsto1.Despitebeingthemostpopularapproximationmethod, P KG ( R ) canbeonlyagood approximationforlongchainsthatarenotstretched.Therefore,itisnotsuitableformodelsof rubberelasticitytoderiveforceandenergyduetoitslargeerrorinpredictingtheasymptotic behaviorof ln( P exact ) .The W KG hasaconsiderablylargeerrorwhichcanbecomeevenlargerin shorterchains.Forexample,therelativeerrorofstrainenergyresultedfromKGPDFforashort chainwith8segmentsisatleast8timeshigherthanthatofalongchainwith64segments.While manyotherapproximationssuchas P A KG provideslightlymoreaccurateapproximationsthan KG,theyremainunpopularduetotheextremecomplexityoftheirderivatives.Despitethe factthat P SD hasanacceptableaccuracyevenforshortchains,utilizingthisdistributionfunction isalmostunfeasibleduetoitsmathematicalcomplexity. 3.3ApproximationofNon-GaussianDistribution Theaccuracy-complexitytrade-offproblemincurrentPDFapproximationfunctions(seesection 3.2)necessitatetodevelopafamilyofpreciseandsimpleapproximationthatareparticularlyrele- vantforshorterchains.Comparing P KG with P exact fordifferentchainlengthsshowsarepeating errorwhichcanbeconsideredalmostindependentof n .Inviewofthisasamul- tiplicativeerrorfunctions,onecanconsiderallofthepreviousapproximationfunctionssuchas P WG , P SD and P A KG asspecialsub-classesofamasterapproximationfunction ~ P whichcanbe writtenwithrespectto P KG as 28 (a)RelativeerrorofPDF- n =8 (b)Relativeerrorof ln( P n ) - n =8 (c)RelativeerrorofPDF- n =16 (d)Relativeerrorof ln( P n ) - n =16 (e)RelativeerrorofPDF- n =64 (f)Relativeerrorof ln( P n ) - n =64 Figure3.2.ComparisonbetweenGaussian,KG,AmendedKGandEq.3.11distributionfunction withtheexactPDF(RelativeerrorofPDF a,cande andRelativeerrorof ln( P n ) b,dandf )of idealchainswithdifferentlengths a-b n =8 , c-d n =16 and e-f n =64 . 29 ~ P ( r ) ' P KG ( r;n ) ˚ ( t ) ; (3.12) where ˚ ( t ) isamultiplicativecorrectionfunctiontoreducetheerrorof P KG .Here,we hypothesizethat ˚ ( t ) canbechosentocontroltheaccuracy-complexitytradeofffordifferent applications.Inviewofgoodagreementof P WG , P SD and P Y withexactdistributionforshort chains,onecanconclude ˚ ( t ) shouldhavesamepropertiesastheratioofthesedistributionand P KG .AsdiscussedintheSection3.2, P WG , P SD and P Y havealmostsameformulation.Thus amongthem, P Y isselectedtocalculateanestimationof ˚ ( t ) as ˚ ( t ) ' P Y P KG = 2 t 1 sinh( ) 2 1 2 : (3.13) Thefeatureofthisestimationisitslimitwhen t approachesto 1 ,whichtendsto. ByconsideringtheorderpoleofILFat t =1 [59]and lim t ! 1 1 sinh( ) 2 =1 ,onecan concludethat ˚ ( t ) hassecondorderpoleatthispointanditsresiduecanbecalculatedas R ( ˚ )=lim t ! 1 ( t 1) 2 ˚ ( t )=1 : (3.14) By ˚ ( t ) anapproximation ~ P withgoodaccuracywithrespectto P exact canbeobtained.In thisregard,theapproximationfunctionshouldhavesamepropertiesas ˚ ( t ) ,secondorderpole withresidueof 1 .Thesimplestfunctionwithsecondorderpoleis (1 t ) 2 .Inviewofthesecond orderpoleof ˚ ( t ) and (1 t ) 2 ,itcanbewrittenasarationalfunctionsuchas, ˚ ( t )= a ( t ) (1 t ) 2 ; (3.15) where a ( t ) isapproximationfunctionwhichisusedtoadjusttheapproximationfunctionwiththe exactdistribution.Therearedifferentalternativeformsforestimationsof a ( t ) suchaspolynomial, exponentialandetc..Inthisstudyexponentialfunction( exp[ P m i =1 a i t 2 i 1 ] ,where m isthenumber 30 Table3.1.Relativeerrorofapproximateddistributionfunctionbyconsidering1and2termsfor chainswithdifferentlength. max.relativeerror[%] mn =8 n =16 n =32 n =64 KG100100100100 1118910 23455 oftermsusedintheapproximation)isselectedtoapproximate a ( t ) ,whichwillresultsimplestrain energyfunctions.Inordertoobtainbestapproximationwithleastmaximumrelativeerrorin wholedomain [0 1] ,min maxsolver(fminimax)inMATLABisusedtominimize themaximumrelativeerrorof ~ P ( r ) respecttoexactdistributionfunction.Thecoefof approximationfunctionwithoneandtwotermsisobtainedas P m =1 approx ( r )= P KG ( r;n ) exp( 1 : 75 t ) (1 t ) 2 (3.16a) P m =2 approx ( r )= P KG ( r;n ) exp( 2 t +0 : 29 t 3 ) (1 t ) 2 : (3.16b) Themaxrelativeerrorsoftheseapproximationsforchainswithdifferentlengtharepresentedin Table3.1andsummarizedinFig.3.3.TheyillustrategoodagreementwiththeexactPDFfor chainswithdifferentlengthsinthewholerangeof t . (a) (b) Figure3.3.TherelativeerrorofapproximateddistributionfunctionrespecttotheexactPDFfor chainswithdifferentnumberofsegmenta) m =1 andb) m =2 . 31 3.4EntropicForceofaSingleChain Inpolymerphysics,theelasticretractionforceofasinglepolymerchainisassociatedwithchanges intheentropyofthechainsinthecourseofdeformation.Accordingly,thestrainenergy W = TS ofasinglechainiscalculatedthroughBoltzmann'sentropyrelation, S = k ln( P ( r )) ,where S istheentropyofthechain, T theabsolutetemperature,and k theBoltzmannconstant.Thus,the entropicforce, f n ( r ) ,requiredtoperturbthechainsend-to-enddistanceisgivenby f n ( r )= @W ( r ) @r = kT @ ln( P n ( r )) @r : (3.17) InviewofthecomplicatedformulationoftheexactPDF,theapproximatesareoftenusedtode- scribetheforceofthechaininthecourseofdeformation.Thesimplestapproachistoderivethe forcebasedontheGaussianPDF P G (Eq.3.4)whichyieldstheforceasalinearfunctionofde- formation( f G ( t )= kT l t ).However, P G isvalidforlongchainsandatsmalldeformationregimes, only.Inlargedeformations, P KG isthemostpopularapproximationfunction,whichyieldsthe followingequationforpolymerforce f KG ( t )= kT l : (3.18) Otherapproximationsoftheforcecanbesimplyderivedbyimplementinganyoftheaforemen- tionedPDFapproximationsintoEq.3.17.Forexampleusingthe P SD (Eq.3.11),theentropic forcecanbeestimatedas f SD ( n;r )= kT l ( + 1 n 1 t 2 t ( )+2 5 4 ( ) 2 !) ; (3.19) where = 2 t 1 t 2 .SimilartoPDFs,thecomplexityofaccurateapproximationssuchasEq.3.19 preventthemfromwideacceptance(e.g.compareEq.3.18withEq.3.19). Theentropicforcederivedbasedon P G , P KG , P A KG and P SD arecomparedwiththeforceof theexactPDFandshowninFig.3.4- a and b forshortandlongchains,respectively.Asillustrated 32 inFig.3.4- a ,theforceresultedfromtheKGhaslargerelativeerrorswithrespecttotheexact entropicforceforasmallchain.Inlongerchains,theKGforcehasgoodagreementwiththeexact one.Furthermore,theforceassociatedwith P SD hasthebestagreementwiththeexactentropic force(seeFig.3.4- a-b ).TheFig.3.4- c-d showthattherelativeerrorofthesteepestdecent approximationistheminimuminbothshortandlongchains. (a) (b) (c) (d) Figure3.4.Comparisonoftheentropicforceofasinglechainresultedfromexactnon-Gaussian distributionfunctionanditsapproximations,steepestdecentapproximation,andinverseLangevin function(a-b),andtheirrelativeerrorswithrespecttotheexactfunction(c-d)foraandc) n =8 andbandd) n =64 . 3.5ApproximationoftheEntropicForce Anewapproximationfortheforceofachainisdevelopedbasedonthefollowingobservation.The oftherelativeerror, E n ,oftheforcederivedby P KG isalmostidenticalforthechainswith 33 differentlengths, n .AsshowninFig.3.5,therelativeerrorcanbenormalizedby 1 n , e ( t )= nE n , thevalueofwhichisvaryingapproximatelybetween 100% and 220% ( 5% ).Accordingly,anew approximationfunctioncanbederivedsimplybymultiplyingacorrectionfunction, 1 1 E n ,into f KG .Then,byexpandingthecorrectionfunctionusingthegeometricseriesandusingthetwo terms,theapproximationofexactforceiswrittenas E n = f exact f KG f exact = ) f exact = f KG 1 1 E n ' f KG n X i =0 ( E n ) i ' f KG (1+ E n ) : (3.20) Byreplacingtheerror, E n ,byitsnormalizedvalue e ( t ) n ,theproposedfunctionbecomes f approx ( n;R )= kT l L 1 ( t ) 1 e ( t ) n ; (3.21) Theapproximationcanbeoptimizedbyamoregeneralfunctionforthenormalizederror showninFig.3.5,whichcanmaketheapproximationtoocomplexforpracticalapplications.Here, e ( t ) canbeestimatedthroughaprocedureofapolynomialwithdegreeof m (seeFig.3.5), whichyields Ł Order0 :using e ( t )=1 reducestheEq.19into f m =0 approx ( n;R )= kT l L 1 ( t ) n 1 n ; (3.22) whichhasarelativeerrorvaryingfrom 0% to 120 n % asshowninFig.3.5.Therelative errorof f m =0 approx isaroundhalfoftherelativeerrorof f KG .Inanotherstudy,Horganand Saccomandi[65,66]andlaterBeaty[67]derivedalmostthesameformulationasEq.3.22 bycomparinganestimationofnon-Gaussiantheorywiththeaveragedstretchinthemacro- scopiclevel( = 2 t 1 t 2 )withGentphenomenologicalmodel.Interestingly,itcanbeshown thatGentmodelhasbetteragreementwithnon-GaussiantheorythanKGforshortchains. Thissimplecanstronglyimprovetheconstitutivemodels[68,21,69],thatuse aprobabilityofchainsexistencewithdifferentlengths.Inthesemodels,theforceofthe 34 (a) (b) Figure3.5.TherelativeerroroftheILFa)withrespecttothelengthofchainandb)Averaged normalizedforalllengths. matrixisdeterminedbysumminguptheforcesofchainswithdifferentlengths. Ł Order2 :atwotermpolynomial, e ( t )=1+ t i ,isusedtorepresent e ( t ) .Byminimizingthe approximationerror,thesecondorderpolynomial( i =2 )isselectedinthisstudyduetoits simplicityandlowerrelativeerrors.Theproposedapproximationcanbewrittenas f m =2 approx ( n;R )= kT l L 1 ( t ) 1 1+ t 2 n : (3.23) Thisapproximationhasextremelyhighaccuracy,comparabletothatof f SD (Eq.3.19),as showninFig.3.6.Themaximumrelativeerroroftheproposedapproximationfor n =8 isequalto 1 : 7% ,whichislowerthan 33% errorofthe f KG .Inadditionthe maximumrelativeerrorofproposedentropicforces(Eq.3.22and3.23)forthechainswith differentlengthsareplottedincomparisonwiththemaximumrelativeerrorofLangevin entropicforceinFig3.7.ItcanbeseeninFig3.7thatthemaximumrelativeerrorofKG entropicforcewith 40 segmentisabout5%.Consideringthislimitasanerrortolerance fortheapproximationofNon-Gaussianentropicforce,theproposedsimpleof entropicforceisvalidforthechainswith 4 segments.Notethatbothproposedapproximation functionscanbeeasilyimplementedinmostofthecurrentelasticitymodelsbyreplacingthe 35 inverseLangevinfunction. Inmostphysical-basedmodelsofrubberelasticity,breakageofthechainoccurswhenthe chainsdeformationexceedstheirallowedextensibilitylimit,whichisdeterminedbystrengthof C-Cbonds.Therefore,itisimportantthatmodelsuseanacceptablepredictionoftheforceand theenergyathighextensibilityratios,particularlywhen t isapproaching 1 .Whileforlongchains therearefewmodelstoprovideforceandenergywithenoughaccuracyaround t =1 ,forshort chainsnomodelexiststhatcanaccuratelypredictforcearound t =1 . (a) (b) Figure3.6.Therelativeerroroftheproposedentropicforcefora) n =8 andb) n =64 alongwith therelativeerroroffullsteepestdecentapproximation. Figure3.7.Themaximumrelativeerroroftheproposedentropicforceforchainswithdifferent lengthalongwiththemaximumrelativeerrorofKGapproximation. 36 Table3.2.Max.relativeerrorofapproximateddistributionfunctionsforchainswithdifferent length. max.relativeerror[%] mn =8 n =16 n =32 n =64 KG261363 1(Eq.3.24a)1.20.60.450.3 2(Eq.3.24b)0.280.250.20.14 (Eq.3.26)10.680.650.67 3.6ApproximationoftheEntropicEnergy Inviewoftheproposedapproximationsforthedistributionfunctionandtheentropicforces,one canderiveasetofapproximationsforthestrainenergyfunction.Accordingly,inviewofthePDFs derivedinEq.3.16aand3.16b,thestrainenergy, W ,canbeobtainedthrough W = kT ln( P ( r )) as W m =1 P app ( r )= W KG ( r;n )+ kT (1 : 75 t +2ln(1 t )) ; (3.24a) W m =2 P app ( r )= W KG ( r;n )+ kT 2 t 0 : 29 t 3 +2ln(1 t ) : (3.24b) TherelativeerrorsoftheproposedapproximationsofEq.3.24aand3.24b,asshowninFig3.8, aremoreaccurateincomparisontoKGstrainenergyfunction(seeTable3.2).We alsoproposeasecondapproachtoestimatethestrainenergyfunctionsdirectlyfromtheproposed entropicforcesinEq.3.23byintegratingthemover R .Sincedirectintegrationofforceapprox- imationisnotfeasibleduetothecomplexnatureoftheILF,theintegrationiscarriedoutafter replacingtheILFbyitsapproximation.Recently,manyaccurateILFapproximationswithrelative errorlessthan0.1%havebeenproposedintheliterature(e.g.[2,57])andthususingeachofthose, onecanderivethestrainenergyofthechainsfromEq.3.23asfollows W m =2 f app ( n;r )= n Z t 0 kT l ( t ) 1 1+ t 2 n dt (3.25) 37 Forexample,using L 1 ( y )= x 1 x +2 x 8 9 x 2 (maxrelativeerror 1% )[57],theinternalenergycan bewrittenas, W m =2 f app ( n;r )= kT 8 45 t 5 t 4 2 8 n 17 27 t 3 + n 1 2 t 2 ( n 2)[ t +ln(1 t )] + c (3.26) AsshowninFig3.8,therelativeerrorsofEq3.26arelowerthanthatof W KG .The relativeerrorandthecomplexityofstrainenergycanbeeasilyadjustedbyusingsimplerormore accurateILF. (a) (b) (c) (d) Figure3.8.Therelativeerrorsofapproximationsofentropicenergy(Eq.3.24a,3.24band3.26) respecttotheexactentropicenergyforchainswithdifferentnumberofsegmentsa) n =8 ,b) n =16 ,c) n =32 andd) n =64 . 38 3.7Conclusion Currently,theprobabilitydistributionfunction,forceandenergyofapolymerchainismostlyde- rivedbasedontheKuhnandGrünmodel.However,theKGmodelisonlyvalidforlongchains ( n ˛ 40 )andinducesahigherrorasthelengthofchaindecrease[56,39,38].How- ever,longisolatedpolymermoleculesisoftendoesnotexistinreality.Theyareintheinteraction withothermolecules.Intheoryofrubberelasticity,thesegmentbetweentwocross-linkorentan- glementisconsideredasanon-Gaussianchain.Thus,thenetworksmostlycontainshortchains inthepolymerswithhighcross-linking,whichaccountfortheirlimitedextensibility.Whilethere aresomeotherapproximationmodelswithconsiderablyhigheraccuracy,theircomplexnature preventstheirwideimplementationinlarge-scalemodels.Inthiswork,wepresentedageneric approachtoderiveafamilyofapproximationfunctionsfortheprobabilitydistributionfunction, entropicforceandstrainenergyofapolymerchainwithadjustableaccuracyandcomplexitylevel, whicharesummarizedinTable3.3.Weshowthatwithsamelevelofcomplexity,ourproposed functionsareconsiderablymoreaccuratethancurrentfunctions.Particularlyforshortchainsor chainsunderlargedeformations,ourapproximationfunctionsareatleast10timesmoreaccurate thanKGapproximationsandthusareexcellentoptionstoreplacetheminconstitutivemodels. Wehopethattheproposedfamilyofapproximationscanhelpotherresearcherstoimprovethe modelingaccuracyinpolymerphysics.Inadditiontohelpengineerstooptimizetheaccuracy-cost trade-offinlarge-scalesimulationsbyallowingthemtoselecttheapproximationfunctionsbased ontheapplication. Havingafamilyofapproximationswithdifferentaccuracy-complexitywouldbeparticularly helpfulinsomeapplicationswhereonecertainformoftheapproximationfunctioncanre- ducethecomputationalloadsorincreaseaccuracy.Someofapplicationofproposed approximationoftheoryofrubberelasticityincludesbi-modalpolymericnetworks,constrained swelling,stressinducedorientationofthepolymerchainsandetc..TheproposedNon-Gaussian theorycandirectlyaffectthecontributionofshortandlongchainsinabi-modalpolymericnet- works,whichcontainsvariousproportionsofrelativelyshortandlongchainsspeciallyinhigh 39 Table3.3.SummaryoftheproposedapproximatesforPDF,entropicforceandstrainenergyalong withtheirrelativeerrorswithrespecttoexactones. Eq.Formula Max.Rel.Error[%]for n 8163264 PDF 3.8 P KG = c n sinh( ) exp( ) o n 100 3.16a P m =1 approx = P KG h exp( 1 : 75 t ) (1 t ) 2 i 118910 3.16b P m =2 approx = P KG exp ( 2 t +0 : 29 t 3 ) (1 t ) 2 3455 Entropic Force 3.23 f KG = kT l 33.314.276.453.22 3.23 f m =2 approx = kT l 1 1+ t 2 n 1.730.740.350.17 Strain Energy W KG = nkT +ln sinh + c 26.612.25.92.9 3.24a W m =1 P app = W KG + kT (1 : 75 t +2ln(1 t )) 1.20.60.450.3 3.24b W m =2 P app = W KG + kT 2 t 0 : 29 t 3 +2ln(1 t ) 0.280.250.200.14 3.26 W m =2 f app = kT f ( n 2)[ t +ln(1 t )] + 8 t 5 45 t 4 2 8 n 17 27 t 3 + n 1 2 t 2 g + c 10.680.650.67 elongations[70,71,72].Thecontributionoffreeenergyinthetotalchangeofchemicalpotential ofasolventresultingfromswellingofanetworkcontainshortchainscanbeaffectedbycon- sideringmorerealisticnon-GaussiandistributioninsteadofWall Whiteend-to-enddistribution function,whichcannottakingtoaccounttheextensibilitylimitationsandonlyapproximate theexcludedvolumeeffectinthecompactconformationsregion[73,74].TheKGdistribution functionandentropicforceofchainsisusedtodevelopamodeltopredictthestressinducedori- entationofthepolymerchains[75,76].Anotherpossibleapplicationofcurrenttheoryisstudying molecularorientationofpolymerschainduetofastelongationalw. 40 APPENDIX 41 Therandomproblemwasoneoftheinterestingtopicsintheearly20 th century.The randomproblemintroducedbyPearsoninalettertoNature1905[77,60].Hetriedto solvethedistributionofpositionofamosquitoinaforest.Variousdistributionfunctionshavebeen developedintheliteraturebasedontheFourierintegrationoftheproblem,which ispresentedbyRayleighin1919[49].Inordertoreviewthesolutionsofrandom problem,differentapproachispresentedasbelow. 1DRandomWalk In1-Drandomwalkproblem,theprobabilityofarrivingtoapointwithdistance x fromtheorigin by n equalstepcanbewrittenasbinomialdistribution P n ( x )= 1 2 n n ! n + x 2 ! n x 2 ! (3..1) Considering x ˝ N andusingStirling'sformula( a != p 2 ˇa a e a ),theprobabilitydistribution functionwillbetoGaussiandistribution 2 .Themostwell-knownsolutionofend-to-end distancedistributionisexpressedbytheGaussiandistribution,whichistheprobabilityof1-D randomlinksat P n ( x )= 1 p 2 ˇn exp x 2 2 n (3..2) TheeffortsforenhancingtheGaussiantheoryofrubberelasticitytoamoreexacttheory thegeneralityandsimplicity.Thenon-Gaussiantreatmentofrubberelasticityisdevelopedto accountforthelimitingextensibilityofthesinglechain.Thisleadstoamoreaccuratedeformation- forcerelationshipinthewholerangeofend-to-enddistanceuptoitslimitingvalue.Theentropic forceresultedfrom1Drandomwalkdistribution(3..1)canbewrittenas f 1 D ( x )= kT 2 l n + x +2 2 n x +2 2 ; (3..3) 2 UsingGaussianPDFtocalculateentropicstrainenergyisthebasisoftheNeo-Hookeancon- stitutivemodel. 42 where istheDigammafunction.AsitcanbeseeninEq.3..3,theentropicforcehasaasymptotic behavioraround r = n +2 insteadof r = n . 3DRandomWalk TheprobabilitydistributionofanFJCissameas3-Drandomproblem,whichdescribesthe probabilityofachainendingatacertainpointatdistant ! r canbesolvedby3-Drandom problem.Theprobabilityofonestepwithlength l inanarbitrarydirectionisequaltoprobability ofexistenceofapointonasphere, ( r l ) 4 ˇl 2 .ApplyingtheFourierintegrationofthisprobability, " characteristicfunction "ofasinglerandomstepisderivedas sin( ˆl ) ˆl ,where l isthelengthofa segmentinFJCchain(Kuhnlength),and ˆ theFourierintegralparameter.Duetotheindependent natureofbonds, P n ( r ) canbewrittenbythemultiplicationoftheprobabilitiesofeachbond. Consideringanequalprobabilityforallsteps,the" characteristicfunction "ofFJC, A n ( ˆ ) isgiven as A n ( ˆ )= n Y i =1 ( ˆl i ) 1 sin( ˆl i ) l i = l ! A n ( ˆ )= Z exp ( iˆ:r ) P n ( r ) dr = sin( ˆl ) ˆl n (3..4) Inthenextstep,theprobabilityfunctionachaincanbederivedbytakingtheFouriertransform ofcharacteristicfunction,whichderivedbyRayleighbyusingthediscontinuousintegralof Dirichlet[49].HeusedtheinverseFouriertransformationofEq.3..4toderivethenon-Gaussian PDFas P n ( r )= 1 2 ˇ 2 r Z 1 0 ˆsin ( ˆr ) sin ( ˆl ) ˆl n dˆ; (3.1) whichwouldbediftosolveanalyticallyforlargenumberofsteps, n> 10 .Theexactnon- Gaussiandistributionfunctionfor 3 , 4 and 6 stepsarederivedbyRayleighassetsofdiscontinues polynomials[49].TheexactsolutionofFourierintegralofEq.3.1,oftenreferredtoas fiRayleigh exactdistributionfunctionfl ,wasderivedbyTreloar[42]basedonthetheoryofrandom sampling.Later,WangandGuth[52],Nagai[61],andHsiungetal.[62]reachedtothesame 43 expressionwithdifferentmathematicalapproach. P exact n ( r )= 1 2 n +1 ( n 2)! ˇl 2 r k n r l 2 X k =0 ( 1) k n k n 2 k r l n 2 (3.2) 44 CHAPTER4 AGENERALIZEDAPPROACHTOGENERATEOPTIMIZEDAPPROXIMATIONSOF INVERSELANGEVINFUNCTION 4.1Introduction TheinverseLangevinfunction L 1 ( y ) comesfromaapproximationoftheRayleigh exactdistributionfunction[45],aprevalentdistributionfunctioninpolymerphysicsandrubber elasticity[78].Inpolymerphysics,theentropicforceofapolymerchainwithend-to-enddistance r ,whichconsistsof n segmentswithlength l isgivenby f ( n;r )= k B T l L 1 r nl ; (4.1.1) where T denotestheabsolutetemperatureand k B theBoltzmann'sconstant.TheLangevin functionsisby L ( y )=coth( y ) 1 =y .Thestrainenergyfunctionofapolymerchainis thencalculatedbytheintegrationoftheentropicforce(Eq.4.1.1)over r [11],whichyields N;r )= KTn r nl +ln sinh = L 1 r nl : (4.1.2) Asthechainend-to-enddistance, r ,approachesitsmaximalvalue(correspondingtothecon- tourlengthofafullystretchedstraightchain), Nl ,theforcetendsto.Thisimpliesasymp- toticbehaviorof L 1 ( y ) inthevicinityof y =1 . TheentropicforcesofpolymerchainsasderivedinEq.4.1.1,arewell-acceptedandwidely usedinnonlinearelasticityofsoftmaterials(seee.g.[79]).Currently,mostmicro-mechanical 45 modelsofpolymericsystemsrepresentpolymerchainsusingtheapproximationsofanILF,since ILFapproximationscannotbeexpressedinanexplicitform.TheILFapproximationeffortscanbe intothreegroups:(i)powerseriessuchasTaylorseries[80,81,82],(ii)rational(Páde) functions[83,84,11],and(iii)trigonometricfunctions[85,86,39]. (i) TheTaylorexpansionapproach provideshighaccuracywithrelativelylowcomplexityin themajorityofthedomain [0 0 : 95) ,exceptinthevicinityoftheasymptote, y =1 .The 5non-zerocoeffortheTaylorexpansionoftheinverseLangevinfunctionare introducedin[11],as L 1 Taylor ( y )=3 y + 9 5 y 3 + 297 175 y 5 + 1539 785 y 7 + 126117 67375 y 9 + ::: + O ( y n ) : (4.1.3) Later,Itskovetal.[87]derivedasimplerecurrentformulafortheTaylorseriescoef ofinversefunctions,whichgivesthemintermsofBernoullinumbers.Theycalculated500 termsoftheseriesandshowedthatderivationofthehighertermsisonlyvalidifhigherdigits areconsideredinthecalculationofterms.Recently,Ehret[82]presentedanothergeneralized powerserieswhichhasasmallererrorthanTaylorseries.Threetermsofhis seriesshowedrelativeerrorsmallerthanthevetermsoftheTaylorapproximationofthe ILF. (ii) Rationalfunctions arethesecondclassofapproximations,whichusuallyhavehighercom- plexitybutbetteraccuracyaroundtheasymptote.TheCohenapproximationisthemost commonILFapproximationfunction.Itisarationalfunctioncomposedofathirdorder polynomialasthenominatorandasecondorderpolynomialasthedenominator,generally shownby[3/2],withroundedcoef[83].FollowingtheCohenformula,functions basedonPádeapproximationshaveextensivelystudiedandseveralaccuratefunctions(with accuraciescloseto1%)wereproposed,someofwhicharelistedbelow[88,76,89,90]. 46 Ł Treloar[11]: L 1 Treloar ( y ) ˇ 3 y 1 0 : 6 y 2 0 : 2 y 4 0 : 2 y 6 = 3 y (1 y 2 )(1+0 : 4 y 2 +0 : 2 y 4 ) [1/6], Ł Cohen[83]: L 1 Cohen ( y ) ˇ y 3 36 35 y 2 1 33 35 y 2 whichisfurtherroundedto L 1 Cohen ( y ) ˇ y 3 y 2 1 y 2 [3/2], Ł Puso[84]: L 1 Puso ( y ) ˇ 3 y 1 y 3 [1/3], Ł Jedynak[89]: L 1 Jedynak ( y ) ˇ 3 y 2 : 6 y 2 +0 : 7 y 3 (1 y 2 )(1+0 : 1 y ) [3/3], Ł Kroger[90]: L 1 Kroger ( y ) ˇ 3 y (1 y 2 )(1+0 : 5 y 2 ) [1/4]and L 1 Kroger ( y ) ˇ 3 y 0 : 2 y (6 y 2 + y 4 2 y 6 ) 1 y 2 [7/2], Ł DarabiandItskov[88]: L 1 Darabi ( y ) ˇ 3 y 3 y 2 + y 3 1 y [3/1]. In2015,Kroger[90]presentedaninformativetoshowthetrade-offchallengeofdif- ferentPádeapproximationsbyshowingtheiraccuracyagainsttheircomplexity.He complexitybythesumoftheordersofthenominatoranddenominatorofthePádefunction. However,expressingthecomplexityofafunctioninthiswayisnotaconsistentindicator ofcomplexitysincetheexpansionofthefractionalfunctionscanyieldfunctionswithdiffer- entcomplexity.Accordingly,thisgurecannotshowtherelevanceoftheseapproximations, sinceeachofthemcanbeintotwoormoresimplerterms,whichhavedifferent complexities.Forexample,whiletheCohenformulaismorecomplex(order5)thanthe Pusoformula(order4),itsexpandedversion, L 1 Cohen ( y )= 1 1 y 1 1+ y + y ,islesscom- plexthantheexpandedversionofthePusoformula L 1 Puso ( y ) 1 1 y + y 1 1+ y + y 2 .Therefore,it seemsthatusingahigher-orderPádeapproximatesdoesnotnecessarilyleadtomorecomplex functions. (iii) Trigonometricfunctions arethethirdclassofILFapproximationfunctions,whichwere presentedbyBergstromin1999[86].Heproposedapiecewisefunctioncomposedof trigonometricfunctionswhichgivesoneofthelowestmaxrelativeerrors(0.064%)among theavailableapproximations.Howeverthepiecewisenatureofthefunctionmakesitsimple- 47 mentationchallenging.Toovercomethischallenge,furthereffortsonthisclasswerebased onnon-piecewiseequationsasfollows Ł Bergstrom[86]: L 1 Bergstrom ( y ) ˇ 8 > > < > > : 1 : 31446tan(1 : 58986 y )+0 : 91209 y j y j < 0 : 84136 1 sign ( y ) y j y j > 0 : 84136 Ł Keady[85]: L 1 Keady ( y ) ˇ 6 ˇ tan( ˇy 2 ) 1+0 : 4178tan 2 ( ˇy 2 ) 1+0 : 508tan 2 ( ˇy 2 ) Ł Khiêm[39]: L 1 Khiem ( y ) ˇ 1 y ˇ cot( ˇy ) SimilartothePádeapproximations,thesetrigonometricapproximationsalsosufferedfrom thetradeoffbetweencomplexityandaccuracyofthemethod.Keady'sapproachwasvery accurate(maxrelativeerrorlessthan0.3%)buttoocomplicatedforimplementation.Khiêm's formulawasrelativelysimple,however,ithadloweraccuracy(12.3%maxrelativeerror),and hastwoasymptoticpoints(atzeroandone),whichalsoincreasesitscomplexity[39]. Inrecentyears,therehasbeenanincreasinginterestinminimizingtheerroroftheinverse Langevinfunction[91,2,92,93].Severalothermethodswerealsodevelopedtoincreasetheac- curacyofILFapproximations.Nguessongetal.[92]usedatwo-stepoftheCohen formulatominimizetherelativeerrorofeachstep.TheauthorsimprovedtheaccuracyoftheCo- henformulabyaddinganon-integerpowerseriestotheoriginalformulation,inordertoreducethe relativeerrorfrom5%to0.05%.In2017,Petrosyan[93]presentedanewformulafortheinverse Langevinfunctionwiththecombinationofarationalfunctionandatrigonometricfunction.Their approachgives0.18%relativeerror. Here,ageneralizedapproachisproposedtodevelopanewclassofapproximationswithad- justablelevelofaccuracy.Tothisend,thesummationofanasymptoticfunction(themainfunction) withapowerseriesisusedtoestimatetheinverseLangevinfunction.Thus,theproposedapproach ofthisstudyprovidesaseriesofcorrectingtermstothemainasymptoticfunctions.Thecorrecting termshelptoreducetherelativeerror,meaningthataddingmoretermswillresultinfunctions 48 thataremoreaccurate.Theresultsshowthatnewestimationscalculatedbytheproposedmethod outperformsmanycurrentapproximationsinbothaccuracyandcomputationalcost. 4.2ProposedApproach TheinverseLangevinfunction L 1 ( y ) hastwosimplepolesat y = 1 withtheresidue R es ( L ( y )) by R es ( L ( y ))=lim y 1 ( y 1) L 1 ( y )=lim x (coth( x ) 1 x 1) x = 1 : (4.2.1) Thus,themainfunctionfortheILFestimationshouldmeettwoconditions;(i)beanodd functionwithtwosimplepolesat y = 1 and(ii)havearesidueof-1.Whilesomeofthe proposedILFestimationsmetbothconditions(Cohen[83],Kroger[90]and...),manyofthem onlymeetthesecondconditionsincetheyconsideronlyonepole y =+1 (Pasu[84],Darabi[88], ...).However,therearecaseswherethefunctionsdonothaveacorrectresidueand/orpoles.For example,theWarnerapproximation[94]hascorrectpolesbutthewrongresidue,whichleadsto anerrorof50%at y =1 .However,itcanbetoyieldacorrectresidue 2 y 1 y 2 .The versionshowedsmallererror(max35%near y =0 )thantheoriginaloneoverthewhole domain. Inpolymerphysics,theargumentoftheILFfunction, y = r Nl ,isalwaysapositivevalue,since itrepresentsthexibilityofapolymerchain.Thus,theILFestimationsweremostlyoptimized fortheargumentrangesbetweenzeroandone(withonepoleat y =1 ).Inaddition,power series-basedapproximationsdonothaveaphysicalpoleandthuscannotcovertheILFnearthe pole.Evensomerationalfunctionsdonothaveacorrectpoleand/orresidue.Inthefollowing,the proposedapproachisintroducedat3stages 1. SimpliMaclaurinApproach 2. SimpliMaclaurinApproachwithOptimizedLast-Term 3. OptimizedPowerSeries 49 4.2.1Maclaurinapproach TheMaclaurinApproachisasimpleclassoftheproposedapproachconsistsoftwo parts.Thepartisthemainfunctionwithcorrectpolesandresidue,whichcanreducetheerror oftheestimationtozeroat x =1 .Basedontherequiredaccuracy/complexity,manyofthecurrent approximationswithcorrectpolecanbeusedinthisstepasamainfunction.Inthesecondpart, apowerseriesisderivedtominimizetheerrorinducedbythemainfunctioninallofthedomain exceptthepole( y =(0 1) ).Tothisaim,theMaclaurinexpansionof L 1 ( y ) f ( y ) canbeused toreducetheerror. L 1 ( y )= f ( y )+ Taylor L 1 ( y ) f ( y ) (4.2.2) InEq.4.2.2,themainfunction f ( y ) isanoddfunctionwithsimplepolesat y = 1 anda residueof 1 .GiventhatboththeILFandthemainfunctionareoddfunctions,theTaylorexpan- sionoftheerrorfunctionshouldonlycontainoddpowers.Therefore,theproposedestimationof theILFwithdegree n iswrittenas L 1 n ( y )= f ( y )+ n X i =1 a i y 2 i 1 (4.2.3) Toillustratetheproposedapproach,letconsidertwopossiblechoicesforthemainfunction, namelyaWarner-likefunction 2 y 1 y 2 ,andatrigonometricfunction ˇ 2 tan ˇ 2 y .Bothfunctionsare odd,havesimplepolesat y = 1 andaresidueof 1 .Theerrorofbothmainfunctionsaregiven asthepowerseriesinthefollowing.Theresultingestimationsofthisapproachcanbewrittenas thefollowingfortheWarner-likeformula(mainfunction= 2 y 1 y 2 )inEq.4.2.4andthetrigonometric mainfunctioninEq.4.2.5.Inthesetwoseries,thenumberofaddedtermscanbebased onthelevelofaccuracy( O 2 i for i addedterms). L 1 n;Warner ( y )= 2 y 1 y 2 + y y 3 5 53 175 y 5 211 875 y 7 + ::: (4.2.4) 50 L 1 n;Trigo: ( y )= ˇ 2 tan ˇ 2 y + 3 ˇ 2 4 y + 9 5 ˇ 4 48 y 3 + 297 175 ˇ 6 480 y 5 + ::: = ˇ 2 tan ˇ 2 y +0 : 53 y 0 : 23 y 3 0 : 31 y 5 +0 : 24 y 7 ;::: (4.2.5) Theabovetwofunctionshavesimilarbehavior,i.e.theirrelativeerrorisreducedbyadding moretermsinTaylorexpansion(seeFig.4.1andTable4.1).Thetermoftheseriescorrects thelimitoftheestimationaroundzero,whiletheothertermsimprovetheestimationoverthe domain.InviewoftheMaclaurinapproach,manyestimationfunctionsdevelopedin otherstudiesarejustspecialsubsetsofthisapproach.Forexample,byconsideringtheWarner approximation( 2 y 1 y 2 )asthemainfunction,theCohenformula,( y 3 y 2 1 y 2 ),wouldbetheorder expansionusingMaclaurinapproach. Similarly,considering y 1 y j asthemainfunction,someapproximationfunctionscanbederived usingtheproposedgeneralizedscheme.The y 1 y j functionisanoddfunctionwithcorrectpoles andresidue.Inpolymerphysicssince y> 0 alwaysholds,thefunctioncanbedto y 1 y , whichisnotanoddfunction.Thus,thepowerseriesoftheestimationcontainsbothoddand evenpowers(seeEq.4.2.6).Sincethismainfunctionisnotaoddfunction,relativeerrorsofthis estimationdonothavereducingtrend.Forthisreason,theseestimationsneedtobeconsideredby addingoddnumberoftermstoreachlowerrelativeerror(seeFig.4.2andTable4.1) L 1 ( y )= y 1 y +2 y y 2 + 4 5 y 3 y 4 + ::: (4.2.6) Usingtwoterms,theaboveequationwillbetheDarabiformula 3 y 3 y 2 + y 3 1 y = y 1 y +2 y y 2 (4.2.7) whichisrelativelyanaccurateapproximationwitharelativeerrorof2.6%. Similarto y 1 y function y 3 1 y isusedbyPetrosyan[93].Thismainfunctionsdonothavea 51 (a)Eq.4.2.4 (b)Eq.4.2.5 Figure4.1.Relativeerrorofa)Eq.4.2.4andb)Eq.4.2.5withdifferentnumberofterms. Table4.1.MaximumrelativeerrorofEq.4.2.4,4.2.5,4.2.6and4.2.8withdifferentnumberof terms. NumberofaddedtermsEq.4.2.4Eq.4.2.5Eq.4.2.6Eq.4.2.8 14.95.25132.64 22.92.922.67.44 30.90.97.40.83 40.730.730.833.90 52 pointsymmetryandhavepoleonlyat y =1 .Theproposedapproachcanbeappliedtothismain function,whichtheresultispresentedintheEq.4.2.8andFig4.2-b.Petrosyanusedtheterm ofthepowerseriesinEq.4.2.8andadded y 5 5 sin(3 : 5 y ) insteadofthecubicterminordertoreduce relativeerrorfrom0.83%to0.18%. L 1 ( y )= y 3 1 y +3 y + 4 5 y 3 y 4 + 122 175 y 5 + ::: (4.2.8) (a)Eq.4.2.6 (b)Eq.4.2.8 Figure4.2.Relativeerrorofa)Eq.4.2.6andb)Eq.4.2.8withdifferentnumberofterms. 53 4.2.2Maclaurinwithoptimizedlast-term Thenextsteptoincreasetheaccuracyoftheproposedmethodistothecoefofthelast termoftheerrorpowerseries.Accordingly,aftertrimmingtheseriestothedesirednumberof terms,weminimizetheinducederrorbythelastcoef L 1 ( y )= f ( y )+ n 1 X i =1 a i y i + a n y n (4.2.9) Intheproposedapproach,therelativeerroratpoints0and1isalmostzero(consideringa simplepoleat1andacorrectlimitof lim y ! 0 L 1 ( y )=3 y fromtermofTaylorexpansionof ILF.Addingonepointinthesecondhalfofthedomainseemstoberelevantduetotheimportance oftheerrornearthepole.Here,bychoosingthepoint y =0 : 75( L 1 (0 : 75)=4) ,theformulation toadjustthecoefofthelasttermwiththepower n isgivenas a n = L 1 (0 : 75) f (0 : 75) P n 1 i =1 a i 0 : 75 i 0 : 75 n : (4.2.10) Toillustratetheoftheoptimizedlast-termcoeftheapproximationfunctionsof themainfunction 2 y= (1 y 2 ) derivedbythesimMaclaurinapproachwithandwithoutan optimizedlast-termarecomparedinTable4.2.Therelativeerrorhasbeendroppedbymorethan 50%afterusingoptimizedlast-term.Usingtheproposedapproach,theaccuracyofthemainfunc- tionscanbeimprovedtoyieldrelativeerrorsaslowas0.3%throughthesesimplemainfunctions. RelativeerrorsandequationsoftheversionsoftheWarner-likeformulaandTrigonomet- ricformulaforupto4addedtermsarepresentedinTable4.2.Thistableshowsthatthehigher ordersofthefunctionsoftheWarner-likeandTrigonometricformulasshowsimilarac- curacyandbyaddingmoreterms,thiscanimprovetheoriginalestimation.However, thisimprovementisnotasmuchasthatofadding3terms.Theestimationswithmorethan3terms thathaveagoodrelativeerroratthepoint y =0 : 75 ,therefore,thiscannotimprove theaccuracyanymore. 54 Interestingly,byconsideringtheWarnerapproximation( 2 y 1 y 2 )asthemainfunction,thevery recentKrogerformula,( L 1 301 = 3 y y 5 (6 y 2 + y 4 2 y 6 ) 1 y 2 ),wouldbethethirdorderexpansionusinga Maclaurinapproachwithanoptimizedlast-term.Byusinganoptimizedlast-term,the coefoftheCohenformulacanfurtherbeimprovedto 2 y 1 y 2 +0 : 9 y whichreducestherelative errorfrom4.9%to3.4%. Table4.2.versionofEq.4.2.4and4.2.5withdifferentnumberofaddedtermsbyusing singlepointerrorminimization. L 1 ( y ) relative error% L 1 ( y ) relative error% Warner-likeMainFunctions 2 y 1 y 2 + y y 3 5 2.9 2 y 1 y 2 + y 0 : 42 y 3 0.89 2 y 1 y 2 + y y 3 5 53 175 y 5 0.9 2 y 1 y 2 + x y 3 5 0 : 4 y 5 0.28 2 y 1 y 2 + y y 3 5 53 175 y 5 211 875 y 7 0.73 2 y 1 y 2 + y y 3 5 53 175 y 5 0 : 16 y 7 0.41 TrigonometricMainFunctions ˇ 2 tan ˇ 2 y +0 : 53 y 0 : 23 y 3 2.92 ˇ 2 tan ˇ 2 y +0 : 53 y 0 : 45 y 3 0.89 ˇ 2 tan ˇ 2 y +0 : 53 y 0 : 23 y 3 0 : 31 y 5 0.9 ˇ 2 tan ˇ 2 y +0 : 53 y 0 : 23 y 3 0 : 4 y 5 0.3 ˇ 2 tan ˇ 2 y +0 : 53 y 0 : 23 y 3 0 : 31 y 5 0 : 24 y 7 0.73 ˇ 2 tan ˇ 2 y +0 : 53 y 0 : 23 y 3 0 : 31 y 5 0 : 14 y 7 0.35 4.2.3Optimizedpowerseries SincethenominalerroroftheMaclaurinapproachcannotgetsmallerthan0.3%,inthenextstep, weproposetoallthecoefentsofthepowerseriesinEq.4.2.11including a 0 .Althoughthe 55 residueconditionwouldnotbethisapproachcanprovideasimpleyetaccuraterepresen- tationofILFapproximations.Inthisapproach,itishighlypreferablethatthemainfunctionbean oddfunction,sinceweonlyrequiretheoddcoefofthepowerseries,otherwiseallterms willbeneeded. L 1 n ( y )= 8 > > < > > : a 0 f ( y )+ P n i =1 a i y 2 i 1 a 0 f ( y )+ P n i =1 a i y i foroddmainfunction forevenmainfunction (4.2.11) Inthisequation,thecoef a 0 isaddedtoadjusttherelativeerroratpoint y =1 ,which isinducedsincetheresidueofthemainfunctionisnotoneanymoreaftersetting a 0 6 =1 .This coefalongwiththetermofthepolynomial,willhelptore-distributetherelativeerrorin thewholedomain.Thepresenceofthemainfunctionandtheconditionofthecorrectlimitwhen y ! 0 makestherelativeerroroftheapproximationequaltozeroattheboundaries.By a 0 and a 1 ,therelativeerrorcanbeminimizedatthesepointsandreducedthroughoutthedomain. ThegeneralerrorminimizationforILFestimationcanbecalculatedbyusingthemaximumrelative erroroftheapproximationasanobjectivefunction: min a i ˆ max y 2 [0 ; 1) L 1 ( y ) L 1 n ( y;a i ) L 1 ( y ) ˙ ; (4.2.12) where, L 1 ( y ) istheexactILFfunctionand L 1 n ( y;a i ) isitsapproximationwithaorderof n polynomialandafunctionofthecoefof a i .Theminimizationofthemaximumrelative errorinEq.4.2.12willbeperformedoverthedomainof y =[0 1) .Here,optimizationofEq. 4.2.12hasbeencarriedoutnumericallybyevaluatingtherelativeerrorat100equi-distantpoints intheverticalaxisfrom [0 1000) tocover y =(0 0 : 999) .First,therelativeerroriscalculated ateachpointandthenthemaximumoverthewholedomainis Letus L 1 0 asthesimplestILFapproximationwithanasymptoticfunctionofonlyone term.Thefunctioncanbeimprovedbythecoef a 0 .Thus,followingtheconcept oferrorminimization,theWarnerformulacanbeupdatedto 2 : 4 y 1 y 2 ,whichhasamaximumrelative 56 Figure4.3.One-dimensionalvariationofvalueoftheobjectivefunctionrespecttothecoef ofmainfunction( a 0 )fordifferentformulainEq.4.2.13. errorof20%,incomparisonwiththe50%erroroftheoriginalWarnerformulaandthe33.3%error ofthecorrectasymptoticfunction.Applyingasimilarapproachtothetrigonometricfunction, ˇ 2 tan( ˇ 2 y ) ,therevisedapproximationfunction, L 1 0 =1 : 724tan( ˇ 2 y ) ,willhaveonly9.75% relativeerrorincomparisontothe17.75%erroroftheoriginalmodel.Usingthismethod,even thesimplestmainfunction, y 1 y ,canbeupdatedto 1 : 5 y 1 y whichhasarelativeerrorof50%whichis lowerthan66.6%erroroforiginalfunctionand200%errorof L 1 [1 = 1] ( y )= 3 y 1 y ,proposedin[2]. Thevalueoftheobjectivefunctionforthesethreemainfunctionscanbeplottedwithrespectto thevariationof a 0 inFig.4.3.AsshowninFig.4.3theparameter a 0 canbeoptimizedtoyieldthe leastrelativeerror. L 1 0 ( y )= 8 > > > > > > < > > > > > > : 1 : 724tan( ˇ 2 y ) 2 : 4 y 1 y 2 1 : 5 y 1 y e max =9 : 75% e max =20% e max =50% (4.2.13) TheILFapproximationfunctionscanbefurtherminimizedbytheadditionofanother orderpolynomialtothemainfunction,i.e. L 1 0 ( y )= a 0 f ( y )+ a 1 y .Accordingly,theILFapprox- imationisformulatedwithrespecttotwoparameters a 0 and a 1 .Thisapproximationpromotesthis minimizationproblemtoatwo-dimensionalerrorminimization.Thevalueofanobjectivefunc- 57 Figure4.4.RelativeerroroftheILFapproximationfunctionwithrespecttothecoefents a 0 and a 1 forWarner-likeformulainEq.4.2.14. tionforthesethreemainfunctionsforavariationof a 0 and a 1 canbevisualizeasa3-dimensional plot,whichispresentedinFig.4.4forWarner-likeapproximations.Inordertoinvestigatethe minimumvalueoftheobjectivefunctionwithrespecttothecoefthesameprocedureof one-dimensionalerror-minimizationcanbeappliedtothisstate.Themaximumrelativeerrorsof approximationwiththetrigonometric,Warner-like,and y 1 y mainfunctionsarereduced to2.6%,2.4%,and6.1%,respectivelywiththefunctionsgivenbelow L 1 1 ( y )= 8 > > > > > > < > > > > > > : 1 : 53tan( ˇ 2 y )+0 : 52 y 0 : 976( 2 : 4 y 1 y 2 + y )=0 : 976 L 1 Cohen 0 : 94 y 1 y +1 : 874 y = y 2 : 814 1 : 874 y 1 y e max =2 : 6% e max =2 : 4% e max =6 : 1% (4.2.14) Errorminimizationforhigher-orderapproximates( n> 2 )aremorecomplexthan approximations.Themaximumrelativeerrorsassociatedwiththe n =2 approximatesofthe 58 Figure4.5.Minimummaximumrelativeerrorsofdifferentorderpolynomialfor y 1 y mainfunction incomparisonto[2]. trigonometric,Warner-like,and y 1 y mainfunctionsare0.58%,0.57%and0.70%,respectively. Thecoefofthesethreemainfunctions,alongwiththeirhigherorders,arepresentedin theTables4.3-4.5.Therelativeerrorintheerrorminimizationapproachcanbereducedaslow as 0 : 02% forhigherordersofpolynomialfunctions.Ascanbeseenfromtherelativeerrorfor differentapproximatefunctions,itispossibletoreachtoveryaccurateapproximationsofthe inverseLangevinfunctionwithoutaddingcomplexitytotheformulas.Itisworthnotingthatall presentedformulashereconsistofapolynomialseriesandaverysimpleasymptoticmainfunction. Inordertocomparethelevelofaccuracyofthesefunctions,themaximumrelativeerrorresulted from y 1 y isplottedalongwiththeerrorpresentedin[2].Therationalapproximatefunctionsorder of [ n; 1] ,whichispresentedin[2]canbetosummationof y 1 y andapolynomialseries. AsitcanbeseeninFig.4.5,thelevelofrelativeerrorresultedfromgeneralminimizationapproach ofthisstudycanbecomparedwiththesameapproachoferrorminimizationofrationalfunctions recentlydevelopedbyMarchiandArruda[2]. ThemaximumrelativeerroroftheproposedILFapproximationsassociatedwiththemain functionof x 1 x andWarner-likefunctionarepresentedintheFig.4.6.Thisshowsthe 59 Figure4.6.MaximumrelativeerrorsofthedifferentfractionalestimationsoftheILF L 1 ( y )= n i =1 a i y i m j =1 b j y i . improvementsoftheproposedapproximationsinthissectionwithaedorderofthe fractionalfunction.Furthermore,ourproposedapproximationsusingmainfunctionwithonepole isdepictedinthelefthandsideoftheFig.4.6.Thisshowsthattheseapproximationswithlower degreeofcomplexityhavelowerrelativeerrorinallproposedapproximations.Thesecondsetof proposedapproximationswithcorrectsymmetry(Warner-likeapproximations)havelowermax- imumrelativeerrorthanpreviouslyproposedapproximationswithsamenumberofpoles.Itis worthnotingthatallproposedfractionalapproximationsinthisstudycanbewrittenasasumma- tionofoneortwofractions(consideringeachpole, y 1 y ),whichmeansthatincreasing theorderofproposedformulacandecreasetherelativeerrorwithoutchangingcomplexity. Itshouldbenotedthatforerrorsbelow5%,roundingofthemultiplicativecoefhave somemajorconsequences.Mostcoefinthisstudyarepresentedwiththreedigitsofsig- tosavespaceandavoidcomplicationoftheproposedformulae.However,fornumerical implementation,suchareservedoesnotexistanduserscandirectlyimplementtheaccuratecoef- 60 providedintheappendix. Table4.3.Approximationswiththemainfunctionof a 0 y 1 y + P n i =1 a i y i . 4.3Conclusion Here,anoptimizationapproachispresentedthatcanprovidemultipleapproximatesoftheinverse Langevinfunctionwithdifferentdegreesofaccuracyandcomplexity.Theapproximatesarede- rivedbasedonamainfunctionthatshouldmeettwoconditions;(i)beanoddfunctionwithtwo simplepolesat y = 1 and(ii)havearesidueof 1 .Regardlessofwhetherthemainfunctions 61 Table4.4.Approximationswiththemainfunctionof a 0 2 y 1 y 2 + P n i =1 a i y 2 i 1 . arerationaloratrigonometricfunctions,theaccuracyoftheapproximatesimprovesasthepolyno- mialorderincreases.SuchapproachallowsusforthetimetochoosetheILFapproximation forapplicationbasedontherequiredaccuracywithanacceptablecomputationalcosts. WhilethederivedILFapproximationdoesnotchangetheorderofcomplexityofthemainfunc- tions(Warner,Cohenand...),theyarefasterthanthenewlydevelopedhigher-order Padeapproximations(acomplexbutmoreaccuraterationalfunction)[58,92,89,91,93,2].This isanimportantfeatureintheofelastomerphysicswhereILFapproximations thecomputationalcostofthesimulation,whilehighaccuracynearsingularityisneeded todescribedeformation.Havingafamilyofapproximantswouldbeparticularlyhelpful 62 Table4.5.Approximationswiththemainfunctionof a 0 ˇ 2 tan ˇ 2 y + P n i =1 a i y 2 i 1 . insomeapplicationswhereonecertainformoftheapproximationfunctioncansignif- icantlyreducethecomputationalloads.ExamplesofsuchsystemsincludeMacKintoshchains whereapropervariationoftheILFapproximationshouldbechosentoderivefunction f ( x ) in L 1 ( xf ( x ))= f ( x ) ,orthepolymernetworkswheretheresponseofthenetworkismainlyob- tainedthrough R P ( x ) L 1 ( x ) dx . 63 CHAPTER5 MICRO-MECHANICALMODELINGOFTHESTRESSSOFTENINGIN DOUBLE-NETWORKHYDROGELS 5.1Introduction Hydrogelscontainalargeamountofwater(50-99%)andareaclassofhydrophilicpolymersthat areextremelysoftandmostlybio-compatible.Thesefeaturesmakethemagreatcandidatefor manypharmaceuticalandbiologicalapplications,suchasdrugdeliverycarriersmatrixforcell immobilizationofboneregeneration,spinalcordinjuries,cartilageandfatedefects,andsuper absorbents[95,96].Duetothedispersionofcrosslinkingandthestructuralin-homogeneity,con- ventionalhydrogelswhichareusuallycomposedofasinglenetwork(SN)showedapoorstrength level,fragileresponse,andlimitedextensibilityandrecoverability.Therefore,theycouldnotbe usedinload-bearingapplications.Thelackofmechanicalstrengthinmostofnaturalandsyn- thetichydrogelsincomparisonwiththesoftbio-tissuessuchascartilage,liver,tendon,skin,and arterieshasindeedbeenoneofthemainchallengesformaterialscientistsinrecentdecades[97]. Furthermore,swellingofconventionalhydrogelswasprimarilyanappealingtopicratherthantheir mechanicalresponse.Inrecentyears,swelling-deswellinganddiffusion-deformationbehaviorsof hydrogelshavebeenextensivelystudiedbynumerousresearchgroups[98,99]. Severalmethodsofsynthesissuchashomogenizationofstructure,supplementation,and useofbonds,havebeendevelopedtotoughenhydrogels[100,101].Examplesoftough hydrogelsincludeslide-ringhydrogels,tetra-PEGhydrogels,nano-compositegels,anddouble networkhydrogels.Amongdifferenttypesoftoughhydrogelclasses,doublenetworkhydrogels 64 (DNhydrogels)usuallydemonstratethebestmechanicalproperties[102]. DNhydrogelsconsistofhighlycross-linkednetwork(polyelectrolyte)andlooselycross- linkedsecondnetwork[103].Thenetworkhoststhebonds,whereasthesecond networkismostlyresponsiblefortheloadtransfer.Cross-linkingdensityandthemolarratioof thesecondnetworktothenetworkarecriticalfactorsthatcharacterizetheresponseoftough DNhydrogels[102].Intheviewofexperimentaldata,gelpropertiesareoptimizedwhenthemolar ratioisaround20.Bycontrollingtheinteractionsbetweentheandsecondnetworks,twotypes ofDNgels,connective( c- )andtrulyindependent( t- )DNhydrogels,canbedeveloped[104]. c- DNhydrogelshavestronginteractionsbetweenpolymernetworks,whereassuchinteractions arepreventedin t- DNhydrogels.Ontheotherhand, t- DNhydrogelsexhibitbettermechanical responsethan c- DNhydrogelsifthesecondnetworkislooselycross-linked.Consequently,loosely cross-linkedsecondnetworkrequireshighmolecularweighttoensuretheintegrity. DNhydrogelsalsodemonstrateJ-andS-typesofnonlinearbehaviorunderlargedeformations withinelasticfeaturethatissimilartothestresssofteningoftherubbers,generallyreferred toasthefiMullinseffectfl[1,105].Therefore,constitutivemodelingofhydrogelsisoftenpracticed usingtheconceptsthatareoriginallydevelopedforthestudyofrubberelasticity[106,107].The Mullinseffectintheelastomericmaterialshasbeenextensivelystudiedoverthelast70years,and thereexistseveralconstitutivemodelsintheliteraturetodescribethisphenomenon[108,109,110, 38,111,112,21]. ThedamageinDNhydrogelsmayresultfromtheruptureofcross-linking,asnoispresent inaDNhydrogelmatrix[1,113].Therefore,DNhydrogelmodelsareassociatedwiththeinelastic responseofthegeltothenetworkduetoitshighlycross-linkedstructure[113].Thesecond networkisoftenconsideredtobehyper-elasticalthoughitselasticitymodulusgraduallydecreases duetotheformationandpropagationofcracks.Thesecondnetworkbridgescracksofthe network.WangandHong[114]describedtheresponse,damageandyieldingofaDNhydrogel withrespecttotheOgden-Roxburghpseudo-elasticitymodel[115].Later,Zhao[106]proposed amodelforinterpenetratingpolymernetworkswhichdecomposedthepolymermatrixintoshort 65 andlongpolymernetworks.ThemodelwasbasedonArruda-Boyceeightchainmodelandthe networkalterationtheory.Liuetal.[116]proposedamodelforstresssofteningand neckinginstabilityinDNhydrogels.Theyconsideredtheenergyofanetworkinaparticular directionandtheenergyofamixtureofpolymer-solventusingFlory-Hugginstheory.Themodel attributesstresssofteningtothefractureofthenetwork,andhardeningtothehyper-elastic stiffeningofthesecondnetwork.TheirresultsarevforvariousDNhydrogeltypes,and showedagenerallygoodagreementwiththeexperimentaldata.Recently,Luetal.[117]and Luetal.[118]proposedaphenomenologicalmodeltodescribevisco-elasticandMullins-effect behaviorofthetoughgels. Inthiswork,amicro-mechanicalconstitutivemodelisproposedtodescribethenon-linear responseandthestresssofteningofDNhydrogels.Thefundamentalsofstatisticalmechanics ofpolymersarereviewedintheSection5.2.Then,thegeneralizednetworkdecomposition conceptandthecorrespondingstrainenergyfunctionapproachisdiscussedinSection5.4.Finally, section5.7describestheevaluationoftheproposedmodelincomparisontoexperimentaldata. 5.2StatisticalMechanicsofPolymers Thedeformationgradient, F isconsideredasmultiplicationofmechanical F m andswellingparts v 1 3 p I ofdeformation.Intheswellingpart,thecoef v p isvolumefractionofgeltothefresh geland I istheidentitytensor.Inthisstudy,weassumedthatswelling-dryingisisotopicand happenedbeforemechanicalloading.Therefore,thedeformationispurelymechanicalandduring loading-unloadingthevolumeofsampleremainsconstant.Letusdenotethepositionvectorof achaininthereferenceanddeformedby R and r ,andtheirlengthsby R and r , respectively.Inordertoconsidereffectofswelling/dryingofthesampleonconstitutivemodel, isotropicofthesampleimpliesthattheend-to-enddistanceofthechains R islinearly alterwithlength a R .Accordingly,onehas r = F R ;r = v 1 3 p d R = d R p ; (5.2.1) 66 where F denotesthemicro-scaledeformationgradientappliedonachain, R p istheend-to-end distanceofthechainintheswelledstate,and d = p d F m T F m d thestretchinthedirectionof theunitvector d .Notethatonlyend-to-enddistanceofthechainsandnumberofchainsperunit ofvolumechangeduetoswelling-dryingprocess.Hereafter,thefollowingfontstylesareused forscalar X ,vector X ,andsecond-order X .Moreover,theparameterswithabarsignoverthem = l denotetheirnormalizedvaluewithrespecttothesegmentKuhnlength l . 5.2.1Non-Gaussiandistributionfunction Thepolymersnetworksconsistofpolymerchainswithdifferentlength,thataredistributedin differentdirections.Theprobabilityofexistenceofachainwithend-to-enddistance r andcounter length n canbecalculatedthroughthesolutionof3Drandomproblemas[119], P exact n ( r )= 1 2 n +1 ( n 2)! ˇl 2 r k m X k =0 ( 1) k 0 B B @ n k 1 C C A ( n 2 k r ) n 2 ; (5.2.2) where m isequalto n r 2 and l isthesegmentlength.Thestrainenergyofasinglechainbased onNon-Gaussianprobabilitycanbecalculatedasafunctionof r throughBoltzmann'sentropy formulaandthermodynamicbalance, c ( r )= kT ln( P n ( r )) .Inthisrelation, T istheabsolute temperature,and k theBoltzmannconstant. Duetomathematicalcomplexityandpiece-wisenature,theexactNon-GaussianPDFwasnot suitableforpracticalapplications.Therefore,mostcurrentpolymerelasticitymodelsoftenusethe KGdistributionfunction,whichisderivedfromtheorderapproximationoftheRayleigh's exactFourierintegraldistribution[11].ThefreeenergyofasinglechainbasedonNon-Gaussian PDFcanbecalculatedasafunctionof r throughthermodynamicbalanceas, c ( n; r )= nKT +ln sinh + c 0 = nKT Z t 0 dt + c 0 ; = L 1 ( t ) (5.2.3) where L 1 ( t ) istheILF, t = r n istheextensibilityratioand c 0 iscorrelatedwiththenumber 67 ofchainstoeliminatetheenergyinreferencestate.However,severalstudiessuggestedthatKG estimationisonlyvalidforsufcientlylargechains ( n ˛ 40) [56,120],andhaserrors withrespecttotheexactPDFforshortchains.ThenetworkintheDNhydrogelsconsistsofthe shortandbrittlechains.Therefore,theKGmethodcauseerrorsinconstitutivemodeling ofDNhydrogels.Inordertoaddressthisshortcoming,weusedenhancedKGdistributionfunction forshortchains,whichisdevelopedrecently[121]. ^ c ( n; r )= nKT Z t 0 ^ ( t ; n ) dt; ^ = 1 1 t 2 n : (5.2.4) ThebracketintheEq.5.2.4isaddedtotheILFtoreducetherelativeerroroftheKGdistri- butionforshortchains.GiventhatInverseLangevinfunctioncannotbederivedexplicitly,rational approximationfunctionsareusedtorepresentit[57,2,58].Therefore,thishassame complexityastheKGmodel. 5.2.2Doublenetworkhydrogels DNgelsarecomposedoftwodissimilarinterpenetratingpolymernetworksgenerallyreferredto astheandsecondnetworks.Thenetworkisahighlycross-linkedbrittlenetworkwith highnumberofbonds.Thesecondnetworkisastretchablenetworkwithhyper-elastic behavioruptoverylargedeformationranges,whichislooselycross-linked(Fig.5.1).Experiments showthatundertensiledeformation,thenetworkrapidlybreaksduetoitslackofxibility, andthesecondnetworkkeepstheintegrityofthegel. SeveraltypesofpolymercomponentsareusedinDNgels,amongwhichthemostpopular choicesarePAMPS 1 -PAAm 2 gel[105]andalginate-PAAmgel[122].InPAMPS,cross-links areformedbycovalentbondswhileinalginategels,thepolymersarecross-linkedbyionicbonds. Accordingly,differentdamagemechanismsshouldbeconsideredforeachgeltype.Themodel 1 poly(2-acrylamido-2-methyl-propane-sulfonicacid) 2 neutralpolyacrylamide 68 Figure5.1.SchematicviewofstructuralcompositionoftheDNgelsasthesuperpositionofthe andsecondnetworks.Thenetworkisnotxibleandhighlycross-linked.Thesecond networkisaloosenetworkandishighlyxibleduetoitslongpolymerchains. developedhereisvalidforthosegelsinwhichonenetworkishyper-elasticandtheotherisbrittle, suchascovalentDNhydrogelscomposedofPAMPSandPAAmnetworks. TheclassicconstitutivebehaviorofDNgelsunderuni-axialtensionisplottedinFig.5.2 whichexhibitsinelasticfeaturessimilartothoseofelastomers,suchasstress-softening,primary curveandhardening[1].Comparingtorubbers,constitutivebehaviorofgelslacksthree majorfeatures,namely(i)hysteresisafterthecycle,(ii)largepermanentset,and(iii)long- termhealing.InDNhydrogels,nosubstantialrecoveryinthebehaviourofdeformedgelcanbe observedevenaftersomeweeksofrelaxation[1].Afterthecycle,thehysteresisinsubsequent cyclesisgenerallyreferredtoascyclicdamage.AsitcanbeseenintheFig.5.2,cyclicdamageand permanentsetinthehydrogelsaresmallerthanthoseofelastomers[1].Therefore, theeffectofstress-softeningaftercycleisnegligible. AnotherfeatureofDNhydrogelsisthepre-damageinthenetworkintheirreferencestate, whichisinducedbytheirtwo-stepsynthesizingmethod[123,124].Thetwo-steppolymerization methodisusedtoprocesstheDNhydrogels,whichisdevelopedbyGongetal.[125].Thedamage inducedinthesecondstepoftheprocessisschematicallyshowninFig.5.3.Arigidandbrittle networkispolymerizedItwillbethenimmersedandswelledinasolutiontosynthesizethe secondnetwork.Thisswellingcausesanincreaseinend-to-enddistanceofchains,whichresult tobreakageoftheshorterchains.Afterpolymerizationofthesecondnetwork,thegelshrinksto reachtothenewequilibriumstate,whichwillbehereafterreferredasareferencestate.Inorder tomodelthisphenomena,weassumedthatthechainshavetheend-to-enddistance R inthe 69 Figure5.2.ConstitutiveresponseofaDN-hydrogelspecimenunderquasi-staticcompression, whichincludethreetypicalinelasticfeatures,i.e.,stress-softening,primarycurveandhardening. Nohealingbehaviorhasbeenreported.[1]. state.Thedamagecausedintheswellingstageofthenetworkisconsideredsameintheall directions. Inthiswork,theDNgelmatrixisconsideredastheassemblyoftwoindependentmatrices ontopofeachother(seeFig.5.1).Duringdeformation,networkwithshortchainsbreaks down,whilethesecondnetworkremainsintactandtransferstheloadwithinthematrix.Sincethe forcesbetweenthenetworkfragmentsaretransferredbythesecondnetwork,theconstitutive responseofthesystemcanbeschematicallyshownthroughtheassemblydepictedinFig.5.1. 5.3ConstitutiveModel 5.3.1Networkdecomposition ByconsideringelasticdeformationofDNhydrogelasanearlyincompressiblematerial,itsstrain energyfunction N ( C ) canbedecoupledintoisochoricandvolumetricpartsby(seee.g.[32]) N ( C )= M ( C )+ U ( J ) ; (5.3.1) 70 Figure5.3.Theschematicpictureofthetwo-steppolymerizationmethodtoprepareDNhydrogels andthedamagecausedbyswelling. where C denotestherightCauchy-Greentensor, J 2 =det C and C = J 2 3 C .Todescribethe mechanicalresponseofDNgels,theisochoricstrainenergyofthematrixisconsideredasthetotal strainenergiesofthe(1N)andthesecond(2N)networks.Thenetworksareassumedtoact paralleltoeachother(seeFig.5.3).Accordingly,theisochoricstrainenergyofthegelmatrix M canberepresentedas M = m X i =1 i (5.3.2) where i denotesthestrainenergyofthei th networkperunitreferencevolumeofthematerialand m isanumberofsub-networksinthematerial,whichisequalto2forDNhydrogels. 5.3.2Modularplatform Themodelproposedhereisbasedontheconceptofthemodularplatform[68],whichallows aframeworktobebuiltbycouplingseveralnetworkmodels,eachoneofwhichrepresentinga network.Anetworkmodelcandescribemultipledamagemechanism,whereeachnetworkcanbe 71 Figure5.4.Schematicbreakdownofamodularframeworkconsistingoftwonetworkmodelsfor theandsecondnetworkstoillustratetheconceptofthenetworks,andsub-networks.The sub-networkconsistsadistributionofshortschainwithdifferentlengthandaverageend-to-end distance, R andthesecondsub-networkconsideredtobeasetoflongchainswithsamelengthand end-to-enddistance. derivedfromexistingmodels.Here,weproposeanetworkmodelforthenetwork,andimport anotheroneforthesecondone.Thenetworkmodelscanbesubstituted,upgraded,orremoved withouttheintegrityoftheframework.Networkmodelsaredesignedtoreturnstrain energy;scale-transitionwillbebasedonamicro-sphereconcept;andentropicbehaviorisassumed forpolymerchains.Here,weonlyconsiderpermanentdamage,however,thenumberofadd-on modulescanbeincreasedlateronifdifferentinelasticfeaturesneededtobeadded. InFig.5.4,thecompositionofamodularframeworkconsistingoftwonetworkmodelsfor thetandsecondnetworksisdepicted[68,126].Fromthemicro-mechanicalpointofview, permanentdamageisaconsequenceofdecompositionofthenetworkduringwhichseveral chainsaredeactivatedthroughdebonding.Sincethede-bondedchainswillnotreattachbackto thenetwork,thedamagebecomespermanent.Sincepermanentdamageismainlyassociatedtothe network,thesecondnetworkisconsideredasahyper-elasticnetwork. Eachsub-networkissubjectedtoadifferentuni-axialdeformationanddamagehistoriesac- 72 cordingtotheirdirections.Integratingasub-networkinalldirections,theconsequentnetwork modelisdevelopedasa3Drepresentationofthe1Dmodelofsub-networks.Thecontributionof eachnetworkinstrainenergyisregulatedbythemodeltotheexperiments.Here,byas- sumingthehomogeneousdistributionofpolymerchainsinallspatialdirections,themacroscopic energyofeachnetwork, i canbeformulatedas i = 1 A s Z S d W i d d u ; (5.3.3) where A s representsthesurfaceareaofthemicro-sphere S ,and d d u theareaof A s withthenormaldirection d (seeFig.5.4).Theparameter d i representstheenergyofthei th sub- networkindirection d .Theintegrationofthemacroscopicenergyofthe3Dmatrixcanbecarried outnumerically j ˘ = k X i =1 d i j w i ; (5.3.4) where w i aretheweightfactorsassociatedtodifferentspatialdirections d i for i =1 ; 2 ;:::;k . 5.3.3Firstnetwork:Brittlenetworkwithdamage Thenetworkisconsideredasanassemblyofbrittlesub-networkswithshortchains.The breakageoftheshortchainsisconsideredasthemainsourceofdamageinthenetwork.The processinitiateswiththeirreversiblebreakageofshorterchains,andeventuallyinvolveslonger chains.Let ~ N 1 bethetotalnumberofactivechainsperunitofvolumeinthenetworkinall directions.Inanarbitrarydirection d ,thechainswiththerelativeend-to-enddistance R have differentnumberofsegments n (relativelength)describedbyanormaldistribution, P ( n ) ,withthe average n ,andthestandarddeviation ˙ n . P ( n )= P 0 exp ( n n ) 2 2 ˙ 2 n ! (5.3.5) 73 Therefore,theaveragefreeenergyofthenetworkinanarbitrarydirectioncanbecalculated byconsideringtheprobabilityofexistenceofchainswithdifferentlengthinthatdirection.The summationoftheenergiesofthewholesetofchainslengthavailableinthedirection d further yieldsthefreeenergyofthesub-networkinthisdirectionas d 1 N = X D n ~ N 1 c ( n; ) P ( n ) ; (5.3.6) where D n representsthesetofavailablechainlengths.TheDNgelsfullyrecovertotheirref- erenceaftertheloadisremoved.Thus,thereloadingresponsewillbeidenticalto theunloadingresponseuntilthestretchreachesthemaximumstretch.Theloadingresponse alwaysfollowsaprimaryloadcurve.Suchbehaviorsuggeststhatthedamageispermanentand isafunctionofthemaximumstretchinthatdirection, d max .Accordingly,aslongas d max re- mainsunchanged,damagewillnotincreaseinthesubsequent,unloadingandreloadingcycles. Accordingly,thedamagehereisdescribedasaninteractionoftwosimultaneousprocessesof(i) breakageanddebondingofpolymerchainsandcross-linksand(ii)networkrearrangementwithin thenetwork.TheschematicpictureofthedamageinthenetworkisillustratedinFig.5.5. (i)PolymerChainbreakage: Duringtheprimaryloadingtheend-to-enddistanceofsome chainswillreachtheircontourlengthandcannotbefurtherextended.Here,weassumethatthese chainswillbebrokenordebondedfromtheircross-links.Thisprocesstakesplaceonlyduring theprimaryloading.Assumingthebondtobechemical,thechainswillneitherdebondnorheal insubsequentunloadingsandreloadings.Accordingly,thenumberofactivechainsinthe network, ~ N 1 ,isonlyreducedduringtheprimaryloadandthus,onlywithrespecttothemaximal stretchofthenetwork d 1 m inthedirection d ,whichcanbewrittenas d i max =max ˝ 2 ( ;t ] p ; d i ( ˝ ) ; d i 2 V 3 ^j d j =1 : (5.3.7) Then,inviewofabreakageforceofapolymerchain, f b ,onecandeterminetheshortest 74 Figure5.5.StresssofteningintheDNgelsinthecourseofdeformation.(a)schematicviewof differentfragmentsofthenetworkconnectedbythechainsofthesecondnetwork,(b)internal structureofafragmentinthereferencestate,(c)decomposedsubstructureofafragmentinwhich shorterchainsweredebondedduetheapplieddeformation.(d)schematicrepresentationofchain lengthdistribution,whichshowsexistenceofshortchainsas1 st networkandlongchainsas2 nd network. chainsavailableinadeformedsub-networkas f r n f b ! d n min = 1 ˘ d max v 1 3 p R n; (5.3.8) where ˘ = L f b < 1 representtheaveragelimitofextensibility,whichcauseearlybreakageof thechains,andcross-linksandisamaterialparameter.Similarly, f b isthenormalizedbreakage forceofapolymerchain.Furthermore,inviewofthe P ( n ) ,acut-offlength n max = n +4 ˙ n can beintroduce,abovewhich P ( n ) isconsideredtobenegligible.Thisassumptioncanbereleasedin ordertoreachtohigherdeformationforlongchainsonly.Accordingly,thesetofavailablechains inthedirection d canbewrittenas D n d max = ˆ n n min d max n n max ˙ ; (5.3.9) whichthatthematerialbehaviorinthedirection d isbytheloadinghistoryin thatdirection. 75 (ii)ChainRearrangementwithinaFragment: Inthecourseofdeformation,theshortest chainswillbebrokenordisentangledfromlongerones.Thisprocesswillleadtosometerminated chainsthatdonotcontributetotheelasticityofthenetworkanymore,whilesomeofthemwill rearrangeaslongerchainswithhighernumberofsegments,asillustratedinFig.5.5(b).The conceptofchainrearrangementsuggeststhatthedetachmentofchainsdoesnotnecessarilylead toafulllossoftheirentropicenergy,sincesomeofthechainswillremaininthenetworkaspart ofalongermacro-molecules.Thus,weassumethatthetotalnumberofactivesegmentsinthe networkineachdirectionisdecreasedduetodeformation.Thisassumptionyields X D n (1) n ~ N 1 (1) P ( n )= X D n d max n ~ N 1 d max P ( n )+ N broken ; (5.3.10) where N broken isnumberofinactivatedsegmentsinthenetwork.Here,byassumingthat percent ofbrokenchainswillremainactiveinthenetwork,thenumberofbrokensegmentcanbewritten as N broken =(1 ) X D n (1) 6\ D n d max n ~ N 1 (1) P ( n ) ; (5.3.11) where isamaterialparametersthatgovernstherateofenergydissipation.Next,considering that ~ N 1 d max tobeindependentof n ,and ~ N 1 (1) tobeaconstant,onecanwrite ~ N 1 d max = N 1 d max ; ( x )=1+ P D n (1) 6\ D n ( x ) n P ( n ) P D n ( x ) n P ( n ) ; (5.3.12) where N 1 isanumberofchainsinthenetworkbeforepolymerizationofthesecondnetwork, whichisconsideredasamaterialconstant.Finally,substitutingEq.5.3.12inEq.5.3.6,theenergy ofasubnetworkindirection d isobtainedas 76 d 1 N = v p N 1 X D n d max c ( n; d ) P ( n ) ; (5.3.13) 5.3.4Secondnetwork:Hyper-elasticresponse Thesecondnetworkisthesourceofmechanicalintegrityofthegelandplaysakeyroleinthe elasticityofthegel.Here,usingthemodelofMieheetal.[55],werepresentthenetworkasan assemblyof N 2 chainswiththeaveragelengthof n 2 segments.Thus,bygeneralizingtheconcept offullnetworkmodelinrubbers[54],theenergyofthesecondsub-networkindirection d canbe writtenas d 2 N = v p N 2 c n 2 ; d ; (5.3.14) wheretheparameters N 2 ,and n 2 arematerialparameters.Thesecondnetworkisanelasticnetwork withafnemotionofcross-links,whichconsistofidenticalchains.Inastressfreestate,thechains areassumedtobeintheunperturbedstateinwhichthemeanend-to-enddistanceofachainis R 0 = p n 2 .Acomprehensivereviewoftheavailablehyper-elasticmodelsthatcantheplatform isprovidedinMarckmannandVerron[127]. 5.4Macro-scaleResponse 5.4.13Dgeneralization Assumingahomogeneousspatialdistributionofpolymersinthegelmatrix,andinviewofEqs. 5.3.4,5.3.13,and5.3.14,thetotalmacroscopicenergyofathree-dimensionalgelmatrixisgiven as 77 M = 1 N + 2 N = v p N 1 k X i =1 8 > < > : w i d i max X D n ( d i max ) P ( n ) c n; d i 9 > = > ; + v p N 2 k X i =1 w i c n 2 ; d i (5.4.1) Theconstitutiveequationforthefstresstensor P canbewrittenby P = @ M @ F = @ 1 N @ F + @ 2 N @ F ; (5.4.2) where @ 1 N @ F = v p N 1 k X i =1 w i @ d 1 N @ d i 1 2 d i @ d i C d i @ F : @ F @ F ; @ 2 N @ F = v p N 2 k X i =1 N 2 w i @ ( n 2 ;x ) @x x = d i 1 2 d i @ d i C d i @ F : @ F @ F : (5.4.3) Theseequationscanbefurtherbymeansofthefollowingidentities @ c n;xv 1 3 p R @x = v 1 3 p RKT ^ xv 1 3 p R n ;n ! ; (5.4.4) @ d 1 N @ d i = v p N 1 d i max X D n ( d i max ) P ( n ) @ c ( n;x ) @x x = d i (5.4.5) @ d C d @ F : @ F @ F =2 F ( d d ): J 1 3 I =2 J 1 3 F ( d d ) : (5.4.6) IntheEq.5.4.4, ^ istheversionofLangevinelasticforceforashortchain.Thus,Eq. 78 5.4.2yields P = v 2 3 p ^ N 0 k X i =1 ( P 1 N ( d i )+ P 2 N ( d i )) w i d i J 1 3 F ( d i d i ) ! ; (5.4.7) where ^ N 0 isequalto N 1 kTR and P 1 N ( x )= x max X D n ( max ) P ( n ) L 1 ( t ) 1 1+ t 2 n ;t = x v 1 3 p R n P 2 N ( x )= N 2 ^ N 0 p n 2 L 1 x v 1 3 p p n 2 ! : (5.4.8) AproperapproximationapproachfortheinverseLangevinfunctionisrequireddependingonthe elongationrangeofpolymerchains.Inthisstudy,duetothehighelongationratioofthechains (relativelylargevalueof L ( f y ) ),afractionalapproximationwitherroroflessthan0.02%ismore favorable(see[57]).Accordingly L 1 ( x ) ˘ = x 1 x + m X i =1 a i x i : (5.4.9) wherethenumberofterms m =5 andthevaluesof a i aregivenin[57].Moreoverinthisstudy, thenumericalintegrationovertheunitsphereisevaluatedbyusing45integrationpointsoverhalf sphere.Thepresenceofpre-damageinthematerialleadstoastep-wiseyieldingpointsinthe stress-straincurve,whichcanbesmootherbyincreasingthenumberofintegrationpoints(see Fig.5.6-d).Thisnumberofintegrationpointswasfoundtoyieldthebestoptimizationbetween computationalcostsofintegratingoverasphereandtheresultederroroftheinducedanisotropy [128,129]. 79 5.5ThermodynamicConsistency Sincethestrainenergyofthegelmatrix M isbyonlyoneinternalvariable,namely d max ,onecanrewrite M as M = M ( C ; max max max )= ~ M ( F ; max max max )= 1 N ( C ; max max max )+ 2 N ( C ) ; (5.5.1) where max max max = ˆ d max : d 2 V 3 ^j d j =1 ˙ : (5.5.2) ThesecondlawofthermodynamicscanbereducedtotheClausius-Duheminequalitytoshowthe thermodynamicconsistencyofthemodelinanarbitrarydirection d @ D m M _ D m ! 0 8 d : (5.5.3) Themaximumstretchremainsconstantduringunloadingandreloading.Therefore, _ d max =0 in unloading-reloadingwhile _ D max > 0 intheprimaryloading.Thus,satisfactionoftheClausius- Duheminequalityduringtheloadingissuftoprove(5.5.3),asonecanwrite @ M @ d max 0 8 d (5.5.4) Withrespectto(5.3.2)and(5.3.4),equation(5.5.3)yields @ M @ d max = @ d 1 N @ d max 0 8 d : (5.5.5) Withoutlosinggenerality,(5.5.5)canbeprovedforanarbitrarydirection d ofprimaryloading. Forthesakeofbriefness, d max and d arereplacedby x inprimaryloadingandinordertotakethe derivationofsummationsinthemodel,thesummationsarereplacedbytheirequivalentintegration 80 .Using(5.3.13),onecanfurtherobtain @ d 1 N @ d max = @ d 1 N @x = v p N 1 2 4 d ( x ) dx X D n ( x ) c ( n;x ) P ( n ) dn min ( x ) dx ( x ) c ( n min ( x ) ; x ) P ( n min ( x )) 3 5 (5.5.6) where n min ( x )= 1 ˘ xv 1 3 p R , dn min ( x ) dx = 1 ˘ v 1 3 p R ,and d ( x ) dx = R ˘ v 1 3 p P ( n min ( x )) n min ( x ) 1+ x ) P D n ( x ) n P ( n ) : (5.5.7) Bysubstituting(5.5.7)in(5.5.6),onecanobtain @ d 1 N @x = N 1 R ˘ v 2 3 p P ( n min ( x )) 2 6 4 n min ( x ) 1+ x ) P D n ( x ) n P ( n ) X D n ( x ) c ( n;x ) P ( n ) ( x ) c ( n min ( x ) ; x ) 3 7 5 0 (5.5.8) As 1 , N 1 R ˘ v 2 3 p P ( n min ( x )) > 0 ,and x ) > 0 ,(5.5.8)holdsifonlywehavethefollowing inequality n min ( x ) P D n ( x ) n P ( n ) X D n ( x ) c ( n;x ) P ( n ) c ( n min ( x ) ; x ) 0 (5.5.9) (5.5.9)canberewrittenas n min ( x ) X D n ( x ) c ( n;x ) P ( n ) c ( n min ( x ) ; x ) X D n ( x ) n P ( n ) 0 (5.5.10) As n min ( x ) and c ( n min ( x ) ; x ) arenotfunctionsof n ,theycanbemovedinsidethesummation, thus 81 X D n ( x ) P ( n )[ n min ( x ) c ( n;x ) n c ( n min ( x ) ;x )] 0 (5.5.11) As n min ( x ) n andthestrainenergyoftheshortestchainisalwayshigherthantheenergy oftherestofthechains ( c ( n;x ) ˝ c ( n min ( x ) ;x )) ),onecanconclude n min ( x ) c ( n;x ) c ( n min ( x ) ;x ) n 0 forall n 2 D n ( x ) .Whilethebracketintheinequality(5.5.11)isless thanzeroforallchainlengths( n ),theproposedmodelholdstheconditionofthethermodynamic consistency. 5.6Analysis Theproposedmodelhasutilizedtenmaterialparameters,sevenofwhichbelongtothe1 st network ( R;n max ; n ;˙ n ; p ; and ˘ ),twoto2 nd network( N 2 and n 2 ),andoneparameternamely ^ N 0 is asimplemultiplicativescalingfactortodescribetheresponseofthepolymermatrix.Outofthe sevenparametersthatareusedinthenetwork,onlyfour, n ;˙ n ; and ˘ ,shouldbeobtained byusingprocedureandtheotherthree, R;n max ; p ,canbeexplicitlyderivedfromthe experimentaldataandmaterialconditions. Ł Parameter R hasnodirecteffectonthematerialresponseintheequilibriumstate.The maincontributionof R isassociatedtotheprobabilityparameters n / R and ˙ n / R , bothofwhicharewithrespectto R .Thus,itcanbecompletelyneutralizedinthe calculationsbysettingitequaltoaconstant(seeFig5.6-a).AsitcanbeseeninFig5.6- a,normalizedstress-straincurvedoesnotchangebyvariationof R aslongasweconsider n =2 R and ˙ n = R . Ł Parameter v p ,thevolumefractionofthegeltothefreshgel,canbedirectlymeasuredfrom thesamples.Notethatthisparameteraffectstheend-to-enddistance, R .Therefore,the constitutivemodelwillconsidertheeffectwatercontentbyadjustingthenumberofchains andalsobyofthestretchforswelled-driedsample.Asshownin 82 Fig5.6-b,thestress-straincurveissoftenedduetoswellingofthesampleandhardenedas thematerialisdried. Ł Parameter n max isusedtoreducethecomputationalloadbyreducingthesummationbound- ariesofEq.5.4.1.Forchainswithsuflarge n ,thestrainenergyisbecomingso lowthattheircontributioncanbeconsideredzero.Accordingly,onecanintroduceacut- offlengthas n max =max n +4 ˙ n ; 1 : 5 R max abovewhichtheenergyofthechainsare simplyconsideredtobezero.AsshowninFig5.6-c,stress-straincurvedoesnotchange byvariationof n max from30to50.Notethatthesmaller n max ,whichiscomparablewith R max actasalimitingstretchandcauseasymptoticbehaviorinlargerdeformation. Ł Parameter P canbedirectlyderivedfromtheexperimentaldataasthelocationof downturnintheprimaryloadingcurve.AsshowninFig5.6-d, P canbeeasily fromthestress-straincurveandisassociatedwiththepre-stretchappliedonnetworkone duringswellingprocedure.Differentstepsofdamagecanbeclearlyseenin,whichis attributedtothenatureofnumericalintegrationin3Dspace.Eachstepofdamageoccurs whenonepointontheunitspherereachestothestretch.Byaddingnumberof integrationpointsthestress-stretchcurvewillbesmootherandmorerealistic. Next,toinvestigatetheeffectsoftheparametersofthe1 st network,thecontributionofsecond networkisminimizedbyconsidering N 2 = N 1 and n 2 =50 .Fig.5.7showsasummaryof theparametricanalysisof n ;˙ n ; and ˘ ,whereeachgraphrepresentsthechangesinducedby changingoneoftheaforementionedparameterswithrespecttothereferenceset,whichispresented inTable5.1. Ł Parameter n and ˙ n cancontrolthetrendofthedamageinthematerial(see5.7-a-b)by amutualeffectwiththelocationof R withrespecttothem.Asmostoftheinternalenergy inthematerialcomesfromthechainswiththehighextensibility,thestiffnessofmaterial relatestothepercentageofthechainwith n ˇ r .Thus,damagewillincreasewhen R reachesthepeakofprobability, n . 83 (a)Variationof R (b)Effectofswelling-drying (c)Variationof n max (d)Variationof p Figure5.6.Theoftheparameters R;n max ; p onthemechanicalresponseofhydro-gels. Redsolidlinesrepresentthereferencecurve,andthedashedlinesshowthechangesinmaterial responseduetothevariationofthecontrolparameters. Ł Parameter willaffectthesofteningofthematerialasitdecreasesasitshowninFig.5.7-c. Asitcanbeseeninthisthematerialwillexperiencemoresofteningduetomore chainloss(smaller )whilethematerialdeforms. Ł Parameter ˘ hasgreateffectonthematerialstiffnessduringtheloading.Asitcanbeseen inFig.5.7-d,thestiffnessandthetotaldamageofnetworkincreasesbyapproaching ˘ to1.Thisphenomenonhappensduetoasymptoticbehaviorofchainforceinthelarge excitability. Torepresentthesecondnetwork, N 2 and n 2 providesufxibilityforthemodelof 2 N to representhyper-elasticresponse.AsshowninFig.5.8a,theparameter n 2 willchangethelocation ofasymptoteintheresponseofthesecondnetworkandthuscangovernthelocationofup-shiftin 84 Table5.1.Thereferencesetofparametersoftheproposedmodel. N 1 KT Rn max n ˙ n p ˘N 2 KTn 2 10[kPa]5302 R R 10.90.9910[kPa]50 theconstitutivecurves.Theparameter N 2 isamultiplicativescalingfactorfor 2 N andaccordingly willjustamplify/de-amplifytheresponse(seeFig.5.8-b).Duetotheabsenceofdamageinrepose ofthesecondnetwork,thesetwoparametershavenoonthestresssofteningnoranyother damagemechanisms.However,astrongersecondnetworkreducesthepermanentdeformationdue tothehighercontractionforceitprovides. 5.7ModelPredictionsvs.ExperimentalResults Inordertovalidatethepresentedmodel,weusedthedataofWebberetal.[1]onuni-axial behavioroftheDNhydrogel.Inthatstudy,theDNhydrogelsweresynthesizedthroughatwo-step sequentialUVpolymerization[125].Thenetworkwasmadefroma1Maqueoussolution ofAMPS 3 crosslinkedwith4mol%MBAA 4 .Thesecondnetworkwassynthesizedafterward aroundtheswollennetwork(fordetailssee[1]).Adumbbell-shapedspecimenwaselongated uptocertainstretchlevelsof1.4,1.46,1.622,and1.72thenunloaded(onecompleteuniaxial tensioncycle).Eachtestwerepreformedonavirginsample. Followingourpreviousdiscussion,fourmaterialparameterscanbeexplicitlyderivedfrom experimentaldataandmaterialconditions,namely Ł N 0 iscalculatedfromthestressatmaxstretch. Ł R consideredtobeaconstantasitsvariationwillnotchangethestress-stretchbehavior. Ł n max isassumedtobethe max [ n +4 ˙ n ; 1 : 1 R 0 max ] . 3 2-acrylamido-2-methylpropanesulfonicacid 4 N,N'methylenebis(acrylamide) 85 (a)Variationof n (b)Variationof ˙ n (c)Variationof (d)Variationof ˘ Figure5.7.Effectofthefourparameters( n ;˙ n ; and ˘ )ofthe1 st networkonthematerial responseinthecourseofuni-axialtension. Ł p isextractedfromthepointthatthecurvatureoftheloadingcurveischanged. Theremainingsixmaterialparametersweretedusingoneloading-unloadingcycleofthe1.72 stretchamplitudeintensiledirection.Thegoodagreementwithotherload-unloadingcurvesin uni-axialtensionaswellascompressionwasobtainedautomatically.Tothisend,theleastsquare errorfunctionwasminimizedwiththeaidoftheLevenberg-Marquardtalgorithm.Theobtained valuesofthematerialparametersaregiveninTable5.2,whilethecurveisplottedagainst experimentaltestinFig.5.9. Inordertoshowtherelevanceofthepre-damage,thedissipatedenergyineachcycleinthe modeliscomparedwiththeexperimentalvalues.Thedissipatedenergyperunitofvolumeduring 86 (a)Variationof n 2 (b)Contributionof2 nd network n 2 (c)Variationof N 2 (d)Contributionof2 nd network N 2 Figure5.8.Effectofthe2 nd networkparametersonthelargedeformationresponseofuni-axial tension.Redsolidlinesrepresentthereferencecurve,andthedashedlinesshowthechangesin materialresponseduetothevariationofthecontrolparameters. thecycle(Mullinseffect)iscalculatedasfollows, U hys = Z loading P Z unloading P (5.7.1) AsshowninFig.5.9-c,thedissipatedenergyinthecycleof =1 : 28 isalmostnegligiblein comparisontocycleswithhigheramplitudesforwhichdissipatedenergygrowsexponentially with .Thenegligibledissipatedenergyforsmallstretchsuggeststhatthematerialdoesnot showanyprimarydamageuntilcertaindeformationlevelisreached.Thus,theearlydamage duringpreparation(pre-damage)ofthematerialcanbeconsideredasthereasonfortheabsenceof theprimarydamage.Thepre-damageisconsideredtobeidenticalinallspatialdirectionssothe chainsinotherdirectionswillreachtothepre-damagestretchlimitgradually.Suchahomogeneous 87 (a)Thedatausedtothemodel (b)Modelpredictionintension (c)Modelpredictionincompression (d)Evolutionofdissipatedenergy Figure5.9.Comparisonofthenominalstress-stretchcurvesofthemodelandtheexperimentfor theuni-axialtensiletests. distributionofpre-damagethatitcanberelatedtoearlyswellingofthenetwork.The dissipatedenergygrowswithaslowerrateinmid-rangestretchesasthestretchappliedonchains exceedsthepre-stretchvalues. Totesttherelevanceofthepresentedmodel,wecomparedthemodelpredictionsagainstthree othertypesofDNhydrogels,namelytheinorganic/organicDN[130],alginateŒpolyacrylamide hybridgelandPNaAMPS-PAAmDNhydrogel[131].Foreachmaterial,oneloading-unloading cycleinwasselectedandusedforofthematerialparameters.Theparametersforeach materialarederivedandsummarizedinTable5.3.Furthermore,thepredictionofthemodelfor otherloading-unloadingcurvesareshowninFig.5.10. 88 Table5.2.Parametersoftheproposedmodeltotheonecycleoftensiletestperformedona DNgelin[1]. N 1 KT Rn max n ˙ n p ˘N 2 KTn 2 1.992[kPa]4.5101.29 R 0.075 R 1.10.870.9860.8[kPa]85 Table5.3.Parametersoftheproposedmodeltothetensiletestperformedonadifferentset ofDNgelsdata. Ref. N 1 KT Rn max n ˙ n p ˘N 2 KTn 2 [130]174[kPa]2104.31.51.00.950.99412.5[kPa]72 [132]132[kPa]2.3304.654.651.050.780.9883.67[kPa]270 [131]14[kPa]76092.31.070.350.99552[kPa]108 89 (a)Inorganic/organicDN[130] (b)AlginateŒpolyacrylamidehybridgel [132] (c)PNaAMPS-PAAmDNhydrogel[131] Figure5.10.Comparisonofthenominalstress-stretchcurvesofthemodelandtheexperimentfor differentDNgels. 5.8ConcludingRemarks Amicro-mechanicalmodelforthein-elasticconstitutivebehaviorofDNgelswereproposedbased onnetworkdecompositionconcept.Inthenewmodelthenetworkhostsallthedamagemech- anismsandthesecondnetworkisrepresentedasahyper-elasticnetwork.Thenetworkismod- eledbasedonthenetworkevolutionmodelofelastomerswhichistoaccountforgradual decompositionofthenetworkinthecourseofdeformation.Usingadirectionaldescriptionof damage,a3Drepresentationofthenetworkisdevelopedwhichcandescribetheevolutionof damageindifferentdirectionswithrespecttotheapplieddeformation.Here,thedamageis astheresultoftwosimultaneousprocedure,(i)debondingofchains,and(ii)partialsofteningof 90 thefragmentsofthenetwork.Thethermodynamicconsistencyofthemodelisv secondnetworkismodeledbyfull-networkhyper-elasticmodel.Themodelisbench-marked againstseveralsetsofexperimentaldataselectedtorevealthedirectionalsofteningin thematerial,andthemodelshowsgoodagreement.Thisfact,besidesthesimplicityandthere- ducedprocedure,makestheproposedmodelasuitableoptionforcommercialandindustrial applications. 91 CHAPTER6 APHYSICALLY-BASEDMODELOFSTRESSSOFTENINGWITHNECKING PHENOMENAFORDOUBLENETWORKGELS 6.1Introduction Hydrogelsareelastomericgelswithchemical(ionic,hydrogen,orcovalent)and/orphysicalcross- links.Thesematerialsswellinwaterwithoutdissolvingandcancontainwateruptohundredtimes oftheirdriedvolume.Inviewoftheirhighwatercontentandextremelysoftnature,hydrogels areconsideredasrevolutionarymaterialsindifferentsuchasdrugdeliverycarriersmatrix forcellimmobilizationofboneregeneration,spinalcordinjuries,cartilagedefects,andsuper absorbents.Duetopoorstrengthlevelandfragileresponseofconventionalsinglenetworkgels, investigationoftheirmechanicalbehaviorhavenotbeenasubjectofinterest.However,several studieshavebeenconductedontheirswelling-deswellinganddiffusion-deformationbehaviors [98]. Overthelasttwodecades,numerousstrategies,suchastheuseofbondsand supplementation,havebeendevelopedtoimprovethemechanicalpropertiesofhydrogelsandfab- ricatetoughhydrogels,includingNano-compositegelsanddoublenetworkhydrogels[102,133]. Amongdifferenttypesoftoughhydrogels,DNgelshaveattractedincreasingattentionduetotheir uniquemechanicalperformance.TheDNgelshavebeendesignedwithtwosetsofinterpenetrated cross-linkednetworks.Thenetwork,e.g.PAMPS,ishighlycross-linkedandrigidwhilethe secondnetwork,e.g.PAAm,islooselycross-linkedandsoft. Thetougheningresultedfrominteractionsbetweenpolymernetworksshowedim- 92 provementsinmechanicalcharacteristicscomparabletoload-bearingtissues.Thesemechanical propertiesdemonstrategreatapplicationpotentialsofDNgelsinmanyThenon- linearbehaviorofDNhydrogelsunderlargedeformationissimilartothemostofelastomericma- terialsespeciallyrubbers,whichincludeJ-typeandS-typeinelasticloadingand softeningduringtheunloading,knownasfiMullinseffectfl[1,105].However,thebehav- iorofthematerialduringconsecutivereloadingofDNgelsisalmostthesame,whichpointsto negligiblehysteresisinthebehaviorofmaterial.Moreover,DNhydrogelwiththenetwork withlowcross-linkdensityor -radiationformationdemonstratetheneckinginstabilityinlarge deformations. Utilizingthisbehavior,researchersproposedconstitutivemodelsforDNgelsusingthecon- ceptsclosetotheoneswhichhavebeendevelopedforrubberymaterialsoverthelast70years [38,111,78].ExamplesofthesestudiesformullinseffectincludeOgden-Roxburghpseudo- elasticitymodel[115]byWangandHong[114]andthenetworkevolutionmodelby MorovatiandDargazany[48].Inaddition,Zhao[106],RikuandMimura[134]andLiuetal. [116]enhancedtheideaofusingsuperpositionofadamageablenetworkandhyper-elasticnet- worktodescribeneckinginDNgels.Despitethefactthatthesemodelcaneffectivelymodel theneckinginstabilitythroughelementanalysis,mostofthemarenotcomparedagainstthe experimentalneckinginstabilitydata. Severalstudiesdemonstratethatthehighlycross-linkednetwork,whichcontainsshort chains,isresponsiblefordamageinDNhydrogels[1,113].Breakageoftheshortchainsinthe networkdevelopsnumerousclustersofhardnetwork,whichactlikeintherubber. Inaddition,thesecondnetworkisaccountableforpreservationoftheintegrityofthefractured networkandhardeningofthematerialinrelativelylargedeformations[113].Although,recent advancesinthemodellingofgelshaveshownimprovementsintheconstitutiverelation ofDNgels,ourunderstandingofloadtransfermechanismwithinthenetworkshasremainedsparse andinconclusive.Inaddition,clustersofnetworksplitintosmallandstifferpiecesduringthe necking,whichactascross-linkforthesecondnetwork[135]. 93 Descriptionofdamageprocessandmodelingofhighfractureenergy,stresssofteningandneck- inginstabilityoftheDNgelisacriticalissuefortheapplicabilityofDNgels[135].Inthisstudy, acontinuumscaleconstitutivemodelofDNgelsinquasi-staticdeformationisdevelopedwhich canbeparticularlyusedtoelucidatetheinelasticfeaturessuchasstresssoftening-hardeningof materialduringprimaryloadingandpermanentdamageduringdeformation.Themodelattributes stresssofteningtodamageinthenetwork,andhardeningtothehyper-elasticstiffeningofthe secondnetwork.Inaddition,irreversibledisentanglementofshortchainsofthesecondnetwork fromnetworkclusterdeterminesthestresssofteningduringneckingstage.Thepaperisorga- nizedasfollows.First,thestatisticalmechanicsofshortchainsarereviewed.Then,generalized networkdecompositionconceptandthecorrespondingstrainenergyfunctionofeachnetworkare discussed.Finally,theevaluationoftheproposedmodelagainstexperimentaldataandthecon- cludingremarkarepresented. 6.2DoubleNetworkPolymers DNgelsareconsistoftwopolymernetworksinwhichthesecondnetworkisformedinthepresence ofthenetwork.DNgelscanbeconsideredasaninterpenetratingpolymernetworkinwhich multiplenetworksareinterwovenandentangledtoeachother.However,thesetwonetworksare notcovalentlybondedtotheothernetworkschain.Aschematicviewofthenetworkandthe secondnetworkisdepictedinFig.6.1.Thenetworkhasahighcross-linkdensity,which resultsinanetworkwithshortchains.Theshortchainsofthenetworkhavearelatively highinitialextensibilityratio,whichmakesthembrittleandmorepronetomoredamage.The secondnetworkissynthesizedinthepresenceofthenetworkwithlowcross-linkdensity. Thechainsinthisnetworkcanbedividedintotwosub-networks,onewithfreechainsandthe otheroneentangledtochainsofthenetwork.Thefreechainsinthesecondnetworkactas ahyper-elasticstretchablenetworkwiththeabilitytostretchtoverylargedeformationranges. Experimentalevidenceshowsthatthechainsinthenetworkrapidlybreaksinthecourseof deformation,andtheentangledchainsofthesecondnetworkwillbeunzippedandformlonger 94 chains.Thesedisentanglementcanbethesourceofthegelintegrityinlargerdeformation. Here,theDNgelmatrixisconsideredastwomatricesontopofeachother,inwhichsome ofthesecondnetworkchainsentangledtothenetworkchainsfragments(seeFig.6.1).The breakageoftheshorterschainsinthenetworkdecomposesthenetworkintosmallindi- vidualfragments,whilethesecondnetworkchainsconnectthefragmentstoensuretheintegrityof thematerial.Damageinthematrixisrelatedtobreakageofthenetworkchainsanddisentan- glementoftheinteractionnetwork,meanwhile,thesecondnetworkchainsexhibitahyperelastic behaviorwithoutdamage.Asthesecondnetworkchainsintheinteractionnetworkunzipdueto thebreakageofnetworkchains,theytransfertheforcesbetweenthenetworkfragments andareresponsiblefortheinstabilityinthelargerdeformations.Theconstitutiveresponseand statusoftheinteractionnetworkareschematicallydepictedinFig.6.2. Figure6.1.SchematicsofaDNgelnetwork,decompositionofitsnetworksandtheirchaindis- tributions.Thehighlycross-linkednetworkwithshortandbrittlechainsandloosesecond networkwithlongandxiblepolymerchains. 6.2.1Neckinginstability Polymerformationstructuree.g.degreeofpolymerizationandcross-linkdensity controlthemechanicalresponseofthematerial,whichcancausedifferentnon-linearfeatures 95 suchasstresssofteningandmaterialin-elasticity.Forinstance,theneckinginstabilityisobserved intheDNhydrogelwiththenetworkwithlowcross-linkdensityor -radiationformation [136].Inthisstudy,weaimtoproposeaconstitutivemodelforDNhydrogel,whichexhibitsboth stress-softeningandneckinginelasticbehavior.DeformationprocessintheDNgelwithnecking isschematicallydepictedinFig.6.2.Asitcanbeinthisthestress-stretchbehaviorofthe materialwiththeneckinginstabilitycanbesplittothreeindividualpartincluding (i) Pre-necking Theevolutionofdamageinthepre-neckingstagecanbeconsideredasbreakage ofbrittlechainsofthenetwork.Thematerialexperiencethesofteningthroughoutpre- neckingstageduetothebreakageofthenetwork. (ii) Neckinginstability Thenetworkchainswillbedividedintothesmallclusterconnected withlongchainsofthesecondnetwork,whichprogressofthisdamagewillcausetheneck- ing.Thiscatastrophicbreakagecontinuesuntilthecontributionofthenetworkbecome negligibleandonlyactsasainsidethesecondnetwork.Clusterssplitintosmalland stifferpiecesduringthenecking.Presenceofthenetworkasas-linkplayan importantroleinthestiffnessofthematerialthroughtheneckingstage.Thestiffnessofthe secondnetworkchainsincreasesduetotheincreaseofitscross-linkingdensityanditsbe- haviorchangeswithrespecttothedamageinthenetwork.Theentangledchainsofthe secondnetworkreleasefromthewhichresultsinthesmoothincreasingdeformation withoutfurtherforce,neckingstage.Thereleasedshorterentangledchainsinthesecondnet- workreleaseaconsiderableamountoftheenergy,whichcanberelatedtotheenergyreleased intheneckingstage.Remarkablestresssofteningisobservedduringtheneckingduetothe continuousreleasingthechains[33]. (iii) Hardening Hardeningisthelaststagewherethesuddenincreaseinstiffnessofthematerial isobservedafternecking.Thisstagecausedbylimitingstretch-abilityofthesecondnet- work.chains.Thestretchinthechainsincreasesashighasthelockingstretchofthelong chainsofthesecondnetwork.Thus,thehardeninginthematerialstartstogrow,whenthe 96 (a)Statuesofinteractionnetwork (b)Theconstitutivebehavior Figure6.2.SchematicviewoftheconstitutivebehaviorandchainunzippingofaDNgelunder quasi-statictensionshowinginelasticfeaturessuchasstresssofteningafterloading,permanent damageandneckinginstability. entanglementsbetweentheclustersandthesecondnetworkdisappear.Thesecondnetwork isconsideredtobethemainstresscontributorthroughoutthehardening.Thehardening proceedsuptothecompletefailureofthematerial(seeFig.6.2). 6.2.2Networkdecomposition Thedeformationgradient, F isdecomposedastwoserieseventofmechanical F m anduni-from swelling v 1 3 p I .Theswelledpartshowsthevolumeofthesampleischanged v p timesuniformly duetowaterabsorptionorrelease.Here,weassumedthattheseprocesshappensequentialandthe sampleisnearlyincompressibleduringthemechanicalloading.Here,thechainend-to-endvector inthereferencestateby r 0 anddeformed r ,andtheirlengthsby r 0 and r ,respec- 97 tively.Theeffectofswelling/dryingofthematerialonconstitutivebehaviorcanbeconsideredby linearoftheend-to-enddistanceofthechains R .Thus,onecanwrite r = F m r 0 ;r = d v 1 3 p R; (6.2.1) where d = p d F m T F m d isthestretchalonganarbitrarydirection d .Hereafter,thefollowing fontstylesareusedforscalar X ,vector X ,andsecond-order X . Thedecompositionofmaterialstrainenergyfunction, N ( C ) ,toisochoricandvolumetric partsyieldsto(seee.g.[32]) N ( C )= M ( C )+ U ( J ) ; (6.2.2) where C = F T F istherightCauchy-Greentensor, J = p det C and C = J 2 3 C .Asdiscussed inthesection6.2.1,theconstitutivebehaviorofDNgelsiscomposedofthreedifferentnetworks response.Therefore,theisochoricstrainenergyofthematerialcanbecalculatedbythesummation ofthe(1N),thesecond(2N)networksandtheinteractionnetwork(12)strainenergy.These networksactinparalleltoeachother(seeFig.6.2).Thus,theisochoricstrainenergyofthegel matrix M canbewrittenas M = 1 + 12 + 2 ; (6.2.3) where 1 and 2 arethe1 st and2 nd networksstrainenergiesperunitreferencevolumeofthema- terial,respectively. 12 denotestheisochoricenergyofsecondnetworkchainsentangledwith networkclustersperunitreferencevolume.Here,thenetworkchainsbreakageanddebond- ingareconsideredassourceofpermanentdamage,asthede-bondedchainswillnotreattach backtothenetwork.Thisdamagecanaffectboththenetworkandtheinteractionnetwork micro-mechanically.Thus,adamageablemodelisconsideredtocapturethesetwonetworksanda hyper-elasticbehaviorwithnodamageisconsideredforthesecondnetworktoensureintegrityof 98 thenetwork. Here,themodelisdevelopedthroughthemodularplatformconcept[137,68].Theframework couplesdifferentmodelstorepresenteachofthenetworks.Here,adamageevolutionmodelis usedtocapturethebehaviorofthenetworkandtheinteractionnetwork.Theevolutionmodel adoptedforeachofthesenetworkinthetermofdamagerateandprobabilityofchaindistribution tomodelanetworkwhichisweakeninlargedeformationandagrowingnetworkmodel.Ahyper- elasticnetworkinsertedtotheframeworktocapturethebehaviorofthesecondnetwork.Note thateachofthesenetworkmodelcanbereplaced,upgradedorremovedwithoutaffectingthe generalityofthemodel.Inthisstudy,thenetworkdevelopedtomodelonlypermanentdamage, however,othermodelcanbeaddedtoconsiderotherinelasticfeaturessuchascyclicdamage, progressivedamage,etc. Theentropicenergyofasinglechaincanbecalculatedthroughpolymerstatisticalmechanics. Amicro-spherescale-transitionschemeisdesignedtoestimatethestrainenergyoffull-network model. Thetotalfreeenergydensityfunctionforanetworkcanbeobtainedthroughintegrationof thechainsineachdirection.Theaveragedfreeenergydensityfunctionofthenetworkwitha orientationdistributionfunctionindirection d i , d i C canbewrittenas i = 1 A s Z S d C d i d d u (6.2.4) where S istheunitmicro-sphere, A s isthesurfacearea,and d d u theareaof A s inthe direction d .Inaddition, d j denotesthestrainenergyofthei th sub-networkinthedirection d .The integrationontheunitmicro-sphereofthemacroscopicenergycanbeestimatednumerically i ˘ = k X j =1 d j i w j ; (6.2.5) and w j istheassociatedweightfactortospatialdirections d j .Here,thenumericalschemeis 99 Figure6.3.Schematicframeworkofproposedmodelconsistingofthreesub-networks.The andinteractionnetworksconsideredasastochasticdamageablemodelsandthesecondnetworkas ahyper-elasticnetwork. calculatedthroughthe90integrationpointspresentedin[129].Thisschemeisselectedasaresult oftradeoffbetweennumericalcomputationalcostsandtheresultederroroftheinducedanisotropy [128].Notethattheresultedsummationnumericalschemecanbeinterpretedasaggregationof90 chainsindifferentdirectionandprobability.Inaddition,differentstretchineachdirectionleads toanon-uniformdamageevolutionandhistoriesineachsub-network.3Drepresentationofthe modelcanbeobtainedbasedonthesummationofenergyofeach1Dsub-networkinalldirection. Thus,adamageablenetworkevolutionmodelof1Dsetsofchainsaredevelopedheretomodel constitutivebehaviorofthematerial. 100 6.3Micro-mechanicsofaSingleChain Thefreeenergyofasinglepolymerchaincanbecalculatedthroughthenon-Gaussiandistribu- tionofarandomwalkproblem.Themostprominentapproximationofthenon-Gaussiantheory ofrubberelasticityistheKG.TheKGstrainenergyofasinglechaincanbederivedusingthe combinationoftheBoltzmann˜sentropyrelationandthermodynamicbalanceas, c ( n; r )= nk B T +ln sinh + c 0 = nk B T Z t 0 d˝ + c 0 ; (6.3.1) where k B denotestheBoltzmann'sconstantand T istheabsolutetemperature. ( ) istheILF oftheextensibilityratio, t = r n = L ( )=coth( ) 1 and c 0 ( n ) isaddedtoeliminatethe freeenergyinreferenceInaddition, r denotesnormalizedend-to-endvectorwith respecttothesegmentKuhnlength l .However,KuhnandGrünisderivedtheKGmodelbased ontheassumptionthatthechainsaresuflong,whichtherelativeerrordecreasesasthe chainlengthincreases(about 5% for n =40 )[50,119].Thisapproximationhasasmuchas100 %relativeerrorwithrespecttotheexacttheoryforshortchains(seeFig.).Thus,theKGtheory isnotaproperchoicetodescribeconstitutivebehaviorofshortchainswithhighextensibilityof thenetworkintheDNhydrogels.AnovelandsimpleenhancedKGisdevelopedrecently withremarkableaccuracywithrespecttotheexacttheoryforshortchains,whichcanbewritten as[121] ^ c ( n; r )= nk B T Z t 0 ^ ( ˝ ; n ) d˝; ^ = 1 1+ t 2 n : (6.3.2) Eq.6.3.2asimpletermwithorderof 1 n (thebracket)ismultipliedtotheILFtoimprove theaccuracyoftheenergyfunctionforespeciallytheshortchains.NotethattheILFcannotbe derivedexplicitlyandmostofthemicro-mechanicalmodelsapproximateitwithrationalfunctions 101 Figure6.4.ThemaximumrelativeerroroftheKGandenhancedKGforchainswithdifferent length. orcalculateitimplicitly.Therefore,thishassamecomplexityasKGtheory.In addition,theILFshouldbeapproximatedbyasimpleapproximationwithgoodaccuracyinthe wholerangeofpolymerchainsextensibility[59,2,138].Inthisstudy,theILFisapproximated throughaorderfractionalapproximationwithtwopolynomialterms(relativeerrorof1.0%) as(see[59]) L 1 ( x ) ˘ = 1 1 x + x 8 9 x 2 : (6.3.3) 6.4StatisticalNetworkModel Thepolymericresponseresultedfromtheresponseofindividualchainsandtheirinteractions.In ordertoprovideafulldescriptionoftheconstitutivemodel,westartwiththestatisticaldescription ofthematerialinagivenrepresentativevolumeelement(RVE).Thestatisticalviewleadstoan averageresponseofthematerial,whichcanrepresenttheresponseofthewholeRVE.Forthesake ofsimplicity,weassumedthatallactivechainshavetheiraverageend-to-enddistance,whichis characterizedby r 0 forthenetworkwiththeshorterchainsandGaussianmostprobableend-to- enddistance, sqrtn forthelongerchainsofthesecondnetwork.Thisassumptioncanberelaxed, astheprobabilityoftheexistenceofthechainsisajointdistribution,whichcansimplytake intoaccounttheeffectofstochasticend-to-enddistanceaswellaschainlength.Thetotalstrain 102 energyofthenetworkcanbeevaluatedbyaggregatingthestoredenergyofalltheactivechains intheRVE.Tofurtherinvestigatethemicro-mechanicalchaindistributionforthenon-isolated chains,weproposedanovelandsimplechainlengthdistributionispresentsofthecross-linkage orentanglement,whichwillbedescribedindetailinthenextsection. 6.4.1Probabilityofchainlengthbetweencross-links Polymerchainsarejoinedtogetherduetothepresentofcross-linksandentanglementofthechains withinthepolymermatrix.BothGaussianandnon-Gaussiandistributionfunctionofchainexis- tencewith n segmentisdevelopedbasedontheassumptionofthefullyisolatedchain,which makestheminsufforthedistributionofchainlengthinentangledandcross-linked polymers.Inordertoconsidertheeffectofthepresenceofotherchainsandcross-linkageonthe distributionofchainlength,weassumedthedistributioncanbeobtainedbyconsideringthein- dependentprobabilityofGaussianend-to-endandtheprobabilityofoccurrenceofcross-linkor entanglementinthemiddleofthechainas P n ( n;r 0 )= P G ( n;r 0 ) P cl ( n ) (6.4.1) where P G ( n;r 0 ) istheprobabilitychainexistenceand P cl ( n ) istheprobabilityofthecross- linkage.Theprobabilityoftheexistenceofachainwith n segments(withKuhnlength a )and end-to-enddistance r 0 isassumedtofollow3-DGaussiandistributionas P G = 3 2 ˇna 2 3 2 exp 3 r 2 0 2 na 2 : (6.4.2) In P G ,mostprobablechainhas r 2 0 segmentswhichthattheprobabilityofexistenceof longchainsinthematrixismuchhigherthantheshortchains.Althoughbybestofourknowledge, thephysicsofentanglementandchemistryofcross-linkingistoocomplicatedandisnotknown completely,generallyitcanbeconsideredasarandomphenomena.Here,weassumethatthis randomprocessisuniformandtheprobabilityofthecross-linkingofallsegmentsisthesame.The 103 Figure6.5.Schematicrepresentationofacross-linkednetworkandprobabilityofexistenceof cross-linkattheendofeachsegment. cross-linkingprobabilitycanbecalculatedasaratiooftheconcentrationsofcross-linksagentsto thenumberbackbonesegmentsas p c = 2[ C ] crosslink [ C ] segments (6.4.3) Inthisequation,theconcentrationsofcross-linksagentsismultipliedbytwotoconsiderthateach cross-linkreactswithtwosegments.Thus,theprobabilityofexistenceofachainbetweentwo cross-linkswith n segmentsistheproductoftheprobabilityof n 1 jointswithoutcross-linkand twojointwithcross-link(6.5),whichcanbewrittenas P 1 ( n )= p 2 c (1 p c ) n 1 : (6.4.4) Notethatapolymerchainwithentanglementorcross-linkatamiddleKuhnsegmentsinstead ofendingsegmentsconsideredastwochainswithshorterlengths.CombiningEqs.6.4.2and 6.4.4,onecanobtaintheprobabilityofchainexistencebetweentwocross-linkswith n segment andend-to-enddistance r 0 as P n ( n;r 0 )= P 0 (1 p c ) n 3 2 ˇna 2 3 2 exp 3 r 2 0 2 na 2 ; (6.4.5) where P 0 isanormalizationfactortoensuretheintegrationofprobabilityoverallchainsresults one.Asitcanbeseeninthisequationthechainlengthdistributionfunction, P n ( n;r 0 ) isafunction ofbothend-to-enddistanceandcross-linkingprobability.Increasing r 0 anddecreasing p c both 104 (a)Variationof p c with r 0 =10 (b)Variationof r 0 with p c =0 : 1 Figure6.6.Effectof p c and R 0 ontheprobabilityofpolymerchainexistencewith n segments betweencross-links. leadstoawideprobabilitywithlongerchains.However,theircombinationcanproducedifferent shapeofprobabilityfunctionasitcanbeseeninFig.6.6. 6.4.2Distributionalterationduetodamage TheexperimentalevidenceshowsthatDNgelshavealmostnegligiblehysteresisafterthe cycle,thesameunloadingandreloadingresponse.Thematerialfollowstheprimaryloadcurve afteritreachestothemaximumstretchexperiencedbefore.Basedonthisobservationinaddition tothefactthatdamageispermanent,onecanassumedamageasafunctionofthemaximum stretchinthematerial, max .Thisdamageremainsconstantduringunloading/reloadingcycles untildeformationreachestothe max .However,recentlytheexperimentsbyMaiet.al[139] showedthatDNgelshavedeformation-inducedanisotropy,whichsuggeststhatthedamageisnot uniformindifferentdirections.Thus,weassumethematerialhasauniformdistributioninthe virginstateandthedamagewillbeevolvedinthedirection d i basedonthemaximumstretchin thatdirection.Here,wedescribedthepermanentdamageastwocoincidenteventsofbreakageof thechainsbondsandcreationofthelongerchains.Thesetwoprocessescausethealterationofthe chaindistributionfunctionintwoways(i)changeinthesetsofavailablechainsand(ii)amplifying thepresenceoflongerchains 105 (i) Chainbreakage: Thestrainenergyfunctionofasinglechaintendstoasthechainend-to-enddistance, r ,approachestheultimatelockingvalue nl duringtheprimaryloading.However,achain cannotsustainforceandenergy.So,itwillbreakfromitscross-linkorbondat aneffectivebreakageforce, f e .Thus,thehighlyextendedshorterchainsstarttobreakas theyreachtothelockingstretchsoonerthanlongerchains.Thisprocesscauseaprogressive changeinthechaindistributionasthesetofavailablechainswillshrinkinthecourseof deformation.Byconsideringabreakageforceofachainwith n segment, f b ,thedomain ofavailablechainsinadamagedsub-networkisdeterminedas @ ^ c ( n; r ) @r = k B T l ^ ( t ; n ) f b ! d D n = f d max r 0 ˘ n n max g ; (6.4.6) where ˘ ˇL f b k B T < 1 isthemaximumextensibilityrateanditcontroltheMullinseffect inthematerial.Chainsatthestretchequalto ˘n andforceequalto f b breakfromitsbonds orthecross-link.Similarly,thesetsofavailablechainsfortheinteractionnetworkwiththe mostprobableend-to-enddistance p n canbewrittenas d D n = f max (2 ; d 2 max ˘ 2 ) n n max g ; (6.4.7) wheretheminimumvaluefor n isrestrictedtoavoidchainswithlength1. (ii) Evolutionoflongerchains: Thechainsdisentanglementorbreakagefromthecross-linksleadtoapartiallossofchains entropicenergyupontheirdetachment.Asaresult,somethechainsremainactiveinthe networkandformalongermacro-molecules.Theconceptofalterationofdistributionfunc- tionsuggeststhat fractionofthebrokenchainsactivelycontributeinthenetworkandthe restofwillbedead-end.Adetaileddiscussionofthisconcepthasbeenprovidedin[48]. Thisassumptionyieldstoafactor,whichshowsthatthenetworkcontainmore longerchains(seeFig.)as 106 Figure6.7.Theprobabilitydistributionalterationinthecourseofdeformationandtheschematic viewoflongerchainsactivatedduetodetachmentoftheshorterchains. ~ N i ( max )= N i ( max ) ; ( x )=1+ R D n (1) 6\ D n ( x ) nP n ( n ) dn R D n ( x ) nP n ( n ) dn : (6.4.8) where isaparametergoverningtherateofenergydissipationand N i istheinitialnum- berofchainsinthe i th network,bothofwhichareconsideredasmaterialconstants.Here, weassumedthatallthechainsinthenetworkwillbedeactivated( =0 )andwillnot contributeinthenetwork.Inaddition,duetobreakageofthenetworkchainssomeof interactionnetworkchainswillbeunzippedandformlongerchain.Thus,alloftheinterac- tionnetworkschainswillremainactiveinthenetwork( =0 ).Theseassumptionsaremade heretolimitthenumberofmaterialparametersandcanbereleasedinordertoreachtomore generalformoftheconstitutivemodel. 107 6.4.3Totalenergyofsub-network Thestoredenergyofeachsub-networkdirectioncanbeevaluatedbycalculatingtheaveragevalue offreeenergyoftheallavailablechainsinthatdirection.Accordingtothestatisticaltheoryof elastomerelasticityandthealterationprobabilityofexistenceofchainsinanarbitrarydirection, theaveragefreeenergyofthedamageablenetworksinthatdirectioncanbecalculatedas d i = Z d D n ~ N i d max c ( n; r i 0 d ) P i n ( n ) dn; (6.4.9) where i standsfor(1)andinteractionnetwork(12), N i isthetotalnumberofactivechains intheunitvolumeofthematerialinthe i th networkand d D n thesetofavailablechainlengthsin direction d . r i 0 denotestheend-to-enddistancein i th network,whichis r 0 fornetworkand p n fortheinteractionnetwok.Asthebreakageofthechainsinthenetworkisresponsible fordisentanglementofthechainintheinteractionnetwork,theprocessofchaindebondingisthe sameforbothnetworks.However,networkchainsde-bondduetotheloadtransferbetween twonetworks. 6.5Hyper-elasticModel Thechainsinthesecondnetworkthatdonotentangledbythefragmentsofthenetworkcanbe consideredasfreechainswithhyper-elasticbehavior.Asithasbeenmentioned,thesechainsare responsibleforthemechanicalintegrityofthegel.Letusconsider,thisnetworkasanassemblyof N 2 chainswiththeuniformspatialdistribution.Thisnetworkcontainslongchainswithanaverage lengthof n 2 segmentswiththemostprobableend-to-enddistancesameasinteractionnetwork, whichhassamechainswitharandomentanglementtonetworkchains.Thus,inviewconcept offullnetworkmodel,thestrainenergyofthesub-networksofthesecondnetworkcanbeobtained as d 2 = N 2 c n 2 ; d ; (6.5.1) 108 wheretheparameters N 2 ,and n 2 areconsideredasmaterialparameters.Notethatthishyper-elastic modelcanbeconsideredasentropicenergyofchainsuniformlydistributedoveraunitsphere. 6.6ConstitutiveModel Thetotalmacroscopicenergyofgelmatrixcanbecalculatedthroughtheconceptofnetwork decomposition(Eq.6.2.3),micro-spherescale-transition(Eqs.6.2.4,6.2.5)inandthetotalenergy oftheeachsub-networks(Eqs.6.4.9,6.5.1)as M = 1 + 12 + 2 = N 1 k X i =1 2 6 6 6 4 w i 1 ( d i max ) Z D n ( d i max ) P 1 n ( n ) ^ c ( n;r 0 d i ) dn 3 7 7 7 5 + N 12 k X i =1 2 6 6 6 4 w i 12 ( d i max ) Z D n ( d i max ) P 12 n ( n ) ^ c ( n; p n 2 d i ) dn 3 7 7 7 5 + N 2 k X i =1 w i ^ c ( n 2 ; p n 2 d i ) (6.6.1) Onecanderivetheconstitutiveequationofthefstresstensor P basedontotal macroscopicenergyfunctionas P = @ M @ F = @ 1 @ F + @ 12 @ F + @ 2 @ F ; (6.6.2) where @ i @ F = v p N i k X j =1 w j @ d j i @ d j 1 2 d j @ d j C d j @ F : @ F @ F ;i 2f 1 ; 12 ; 2 g : (6.6.3) 109 ByconsideringtheenhancedKGentropicforce, @ ^ c ( n;r ) @r = KT l ^ ( n;r ) ,onecanfurthersim- plifyEq.6.6.3as @ d j i @ d j = v p N i k B T i d j max Z D n ( d j max ) r i 0 P i n ( n ) ^ n; r i 0 d j dni 2f 1 ; 12 g (6.6.4) @ d j 2 @ d j = v 1 3 p p nk B T ^ xv 1 3 p p n ;n ! ; where ^ = L 1 ( t ) 1 1+ t 2 n .Inaddition,followingidentitiescanbesubstitutedintheEq.6.6.3 @ d C d @ F =2 F ( d d ) ; @ F @ F = J 1 3 I : (6.6.5) Thus,bysubstitutingEqs.6.6.3,6.6.4and6.6.5inEq.6.6.2thePiola-Kirchhoffstresscanbe writtenas P = v 2 3 p k b T k X i =1 ( P 1 ( d i )+ P 12 ( d i )+ P 2 ( d i )) w i d i J 1 3 F ( d i d i ) ; (6.6.6) where P i ( x )= N i i x max Z D i n ( max ) P i n ( n ) L 1 ( t i ) 1 1+ t 2 i n dni 2f 1 ; 12 g ; P 2 ( x )= N 2 p n 2 L 1 x v 1 3 p p n 2 ! ; (6.6.7) where t 1 = x v 1 3 p r 0 n and t 12 = x v 1 3 p p n . 110 6.7FiniteElementLinearization Inthissection,theequationsforthecalculationofsecondPiola-Kirchhoffstressandconsistent tangentmodulus,requiredfortheelementimplementation,arepresented.Theincompress- ibleplanestresssituationintotalLagrangianstrainisconsidered,andindicialnotationis usedinequationsforclarity. InordertoutilizetheNewton-RaphsonMethod,thesecondPiola-Kirchhoffstress, S ,should belinearized.Itsvariationcouldbewrittenas dS = C dE (6.7.1) Where C isthematerialtangentmodulus,and E istheGreen-Lagrangestraintensor.The secondPiola-Kirchhoffstressisas S ij = @ M @E ij pdetC 1 = 2 C 1 ij =2 @ M @ C mn d C mn dC ij pdetC 1 = 2 C 1 ij (6.7.2) ThefourthorderdeviatoricprojectiontensorinLagrangiandescription, d C ij dC kl ,is d C ij dC kl = detC 1 = 3 I ijkl 1 3 C ij C 1 kl (6.7.3) With I asthefourthorderidentitytensor.Bysubstituting(3)into(2),and S n and A 2 @ M @ C ij = S n D i D j (6.7.4) A = D m D n C mn (6.7.5) thesecondPiola-Kirchhoffstressisexpressedas S ij = detC 1 = 3 S n D i D j 1 3 AC 1 ij pdetC 1 = 2 C 1 ij (6.7.6) 111 InordertotheLagrangemultiplier p ,theplanestressassumptionisused,andtheCauchy stressintheout-of-planedirectionissettozero.Inthismanner, p isobtained p = detC 5 = 6 S n F 2 33 D 2 3 1 3 A (6.7.7) Byreplacing(7)into(6),theexpressionfor S ij isobtained S ij = detC 1 = 3 S n D i D j F 2 33 D 2 3 C 1 ij (6.7.8) Thematerialtangentmodulusiscalculatedbytakingderivativeof S withrespectto E C ijkl = @S ij @E kl =2 @S ij @C kl =2 @S ij @ C mn d C mn dC kl (6.7.9) By 4 @ 2 M @ C ij @ C kl = C n D i D j D k D l (6.7.10) Thematerialtangentmodulusiscalculated C ijkl = 2 3 ( detC ) 1 = 3 S n D i D j F 2 33 D 2 3 C 1 ij C 1 kl +( detC ) 2 = 3 C n D i D j F 2 33 D 2 3 C 1 ij D k D l 1 3 AC 1 kl +( detC ) 1 = 3 S n 4 F 33 D 2 3 C 1 ij dF 33 dC kl 2 F 2 33 D 2 3 dC 1 ij dC kl ! (6.7.11) Byknowingthatthesheartermsintheout-of-planedirection, S 31 ;S 13 ;S 32 and S 23 ,arezero, andusingtheVoigtnotation,thematrixformof(1)isexpressedas 112 2 6 6 6 6 6 6 6 6 6 6 4 dS 11 dS 22 dS 12 dS 33 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 C 11 C 12 C 13 C 14 C 21 C 22 C 23 C 24 C 31 C 32 C 33 C 34 C 41 C 42 C 43 C 44 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 dE 11 dE 22 2 dE 12 dE 33 3 7 7 7 7 7 7 7 7 7 7 5 (6.7.12) Intheelementsimulationinplanestresscondition,thedisplacementintheout-of-plane directionisnotanindependentvariable,andtherefore dE 33 shouldberemovedfrom(12).As dS 33 =0 and C 44 =0 ,thestaticcondensationcouldnotbeutilized.Thus,toremove dE 33 ,it shouldbeexpressedintermsof dE 11 , dE 22 and dE 12 .Todoso,andbyhaving E 33 = 1 2 ( C 33 1)= 1 2 F 2 33 1 (6.7.13) dE 33 couldbeexpressedas dE 33 = dE 33 dE dE = dE 33 dF 33 dF 33 dC dC dE dE =2 F 33 dF 33 dC dE =2 F 33 dF 33 dC 11 dE 11 + dF 33 dC 22 dE 22 +2 dF 33 dC 12 dE 12 (6.7.14) Finally,fortheplanestresssituation,(1)couldberewrittenas dS =( C + CC ) dE (6.7.15) where CC =2 F 33 2 6 6 6 6 6 6 4 dF 33 dC 11 C 14 dF 33 dC 22 C 14 2 dF 33 dC 12 C 14 dF 33 dC 11 C 24 dF 33 dC 22 C 24 2 dF 33 dC 12 C 24 dF 33 dC 11 C 34 dF 33 dC 22 C 34 2 dF 33 dC 12 C 34 3 7 7 7 7 7 7 5 (6.7.16) and dF 33 dC ij isgivenby 113 dF 33 dC ij = dF 33 dF mn dF mn dC ij = 1 2 Q mn nj F 1 im = 1 2 F 1 im Q mj (6.7.17) Q = dF 33 dF = F 2 33 2 6 6 6 6 6 6 4 F 22 F 21 0 F 12 F 11 0 000 3 7 7 7 7 7 7 5 (6.7.18) Theterm C + CC isthetangentoperatorintheelementformulation.Thedetailedderiva- tionof S n and C n aregivenintheappendix. 6.8ModelValidation Inthefollowingsection,modelpredictiononMullinseffectandneckinginstabilityiscompared withtheexperimentalresults.Themodel˜smaterialparametersareminimizedinordertosimplify themodel.Inthenetwork,theparameter issetequaltozero,whichmeanstheenergyof allbrokenchainsinthisnetworkwillreleasedandthenetworkwillhavesofteningafter neckinginitiation.Notethatresponsethenetworkisdominantinthepre-neckingandthema- terialparametersofthisnetwork( N 1 , ˘ , p 1 c and r 0 )canbebyusingtheexperimentaldata frompre-neckingstage.Moreover,theprobabilityofentanglementofthesecondnetworkchains tothenetworkclustersisassumedtobeasmallvalue(lessthan5%)andthebreakageofthe clusterdoesnotaffectthenumberofactivechainsintheinteractionnetwork.Thematerialparam- etersoftheinteractionnetwork( p 12 c and N 12 )canbeobtainedbyadjustingmodelinthenecked stageasthisnetworkshouldcancelthesofteningduetobreakageofthenetwork.Finally,the secondnetworkisresponsibleforthehardeningofthematerialinlargedeformationsanditsma- terialparameters( N 2 and n 2 )canbeobtainedfromthehardeningstageoftheexperimentaldata. Notethatthelongestchainavailableforbothnetworkandtheinteractionnetworkisassumed thattobeequalto n 2 asthefreechainsofthesecondnetworkshouldbethelongestchainsinthe matrix. 114 Theproposedmodelhas8materialparametersintotalinwhich N 1 canbeobtainedbyscaling ofthedataandthemodel,the p 12 c ispresettoasmallvalueand6remainingparametersshouldbe obtainedthroughanerrorminimizationapproach.Themaximumdifferenceofthemodelpredic- tionandtheexperimentaldataisconsideredaserrorandamin-maxoptimizationschemeisused tothematerialparameters. Toinvestigatetheperformanceofproposedconstitutivemodel,theexperimentaldataforuni- axialtensiontestsfromtheworkofNakajimaetal.[135]areused.Thetrulyindependent DNhydrogelsusedbyNakajimaet.al.weresynthesizedthroughatwo-stepsequentialphoto- polymerization,wherethenetworkwasmadefroma1MaqueoussolutionofAMPScross- linkedwith4mol%MBAA,andthesecondnetwork,PAMPSwassynthesizedafterwardaround theswollennetwork(fordetailssee[135]).Inordertoinvestigatetheperformanceofthe proposedmodel,itwillbeevaluatedwithasetofuni-axialtensiletests,inwhichvirgindumbbell- shapedspecimenswereelongateduptocertainstretchlevelsashighas11.8,andthenunloadedto 0(onecompleteuni-axialtensioncycle).Throughthemodeltotheloading-unloadingcycle ofatensiontestwiththemaximumstretchamplitudeof11,8materialparametersisadjustedand thecurveisplottedagainstexperimentaltestinFig.6.8. ThemodelcapabilityistestedagainsttwoothertypesofDNhydrogels.First,themodelpredic- tionsarecomparedwiththeexperimentaldataoftensileloading-unloadingforSAPS(1,2,2,9)/AAm (2,0.1,0,97)DNhydrogel.Theloadingbehaviorandtheunloadingbehavioratstretchequalto3, 7.5and13arepresentedintheFig.6.8.Furthermore,thetensilebehaviorofmulti-networkhydro- gelwithinstabilityiscomparedwiththepresentedconstitutivelaw,whichispresentedinFig6.10 alongwiththecontributionofeachnetwork.Asitcanbeseeninthisthenetworkonehas softenningafterneckinginitiationandtheinteractionnetworkgrowtocancelitseffect. Ineachsetsofexperiments,oneloadingandoneunloadingisusedtothematerialparameters. TheparametersforallexperimentaldataarederivedandsummarizedinTable6.1. 115 Figure6.8.Comparisonofthenominalstress-stretchcurvesofthemodelandtheexperimentfor theuniaxialtensileexperiment[3]. 6.9NumericalSimulations InordertofurtherassesstheproposedmodelforconstitutivebehaviorofDNhydrogels,theneck- ingphenomenonismodeledusingtheelementmethod.A 1 1mm 2 specimenisconsidered intheplanestresscondition,andduetothesymmetry,onlyonequarterofthespecimenismod- eledandsymmetricalboundaryconditionsareemployed.Theuni-axialtensiontestisperformed byprescribingdisplacementontherightedgeofthespecimen.Thegeometryandboundarycondi- tionsarepresentedinFig.6.11.Inordertotriggerthelocalization,thelengthoftheleftedgehas beenchosentobeslightlysmallerthantherightedge, l left =(1 )l right ,anditwillbeshown thatbychoosingsmallvalueforthisparameter,theresultsareindependentofthechosenvalue ofthisparameter.However,itshouldbenotedthatastheonsetofneckingintheelement simulationisabifurcationpoint,andtheneckingcouldbecapturedevenby =0 andchoosing loadstepsizebeforethebifurcationpoint. ItshouldbehighlightedthatduetothesofteningoftheDNgel,theboundaryvalueproblem lossesellipticityandbecomesill-posed,andaregularizationmethodshouldbeutilized[140,141, 142].However,thesofteninghasnotbeenregularizedinthissimulation.Inordertodemonstrate oneoftheconsequencesoflossofellipticity,themeshdependence,twotypesofmeshes,namely 116 Table6.1.Materialparametersoftheconstitutivemodeltothetensiletestperformedona differentsetofhydrogelsdata. Experiment N 1 k B Tr 0 ˘p 1 c N 12 k B Tp 12 c N 2 k B Tn 2 [3]29.511.50.9990.195146.606.9236 [4]9.665.50.9990.2723.350.012.21260 [5]190.5150.9970.3314350.015120.450 Figure6.9.Comparisonofthenominalstress-stretchcurvesofthemodelandtheexperimentfor theuniaxialtensileexperiment[4]. alignedandunaligned,withtwolevelsofareutilized,whichareshowninFig.6.12. 4-nodedquadrilateralelementswithfullintegrationareusedinthisstudy. Accordingto[143],thepropagatingneckingcouldbeconsideredasaorderphasetransi- tion,andthetransitionstresscouldbecalculatedaccordingtotheMaxwell'srule.Theexperimen- talandsimulatedstress-stretchcurveswith =1e 3 ,togetherwiththecalculatedMaxwell's linearedepictedinFig.6.13forvariousmeshes.Itcanbeseenthatthemeshdependenceand oscillationsintheneckingpropagationstageoftheelementsimulationaremorepronounced inthecaseofalignedmeshes.Inthiscase,eachoscillationcorrespondstolocalizationofone 117 (a)Pre-necking (b)Necking (c)Contributionofeachsub-network Figure6.10.Themodelpredictionforpre-neckingandfullresponseofthematerialalongwith contributionofeachnetworkintotalresponseof[5]. columnofmeshes,andeachintervalisascaledversionofconstitutivebehaviorfromthepeak tothepointwherethestressreachesthepeakagaininthere-hardening.Therefore,thenumber ofoscillationsisequaltothenumberofcolumnsofelements.Itshouldbehighlightedthatthe sameresultsandconclusionsarepresentedin[144]formodelingoftheLüdersband,whichisa propagatinglocalizationinmetals,verysimilartotheneckingofthehydrogels.Theof ontheresponseoftheneckingforoneofthealignedmeshesispresentedinFig.6.14.Obviously, theresultsareindependentofthechosenvalueforbeta. Inthecaseofunalignedmeshes,thesimulationsandexperimentalresultsareingoodagree- ment;andtheyieldstressandstrain,transitionstressandthehardeningstagearepredictedwith goodaccuracyinthesimulations.Thethreedistinctdeformationstages,namelythepre-necking, 118 Figure6.11.TheelementsimulationoftheneckingofDNhydrogels;geometryandboundary conditions. Figure6.12.TheelementdiscretizationoftheneckingofDNhydrogels;a)Aligned2x2,b) Aligned4x4,c)Unalignedcoarse,d)Unaligned 119 (a)Alignedmeshes (b)UnalignedMeshes Figure6.13.TheNeckingofDNhydrogels:Acomparisonofstress-stretchcurvesbetweenthe experimentalresultsandtheelementsimulation. Figure6.14.TheelementsimulationoftheneckingofDNhydrogels;effectof ,i.e.imper- fectionsize,ontheresponse. 120 Figure6.15.TheelementsimulationoftheneckingofDNhydrogels;variousstagesof necking. neckingandhardeningcouldbedistinguishedinthiscase.Inthepre-neckingstage( < 2 : 4 ), thedeformationishomogeneousandnoneckinghasoccurred.Afterreachingtheyieldstress,the stresshasasuddendropfollowingbyaplateau( 2 : 4 < 4 : 4 ),whichdemonstratethenecking stage.Inthisstage,theneckinginitiatesfromtheleftedgeofthespecimenandpropagatestothe right,andthestresslevelremainsapproximatelyconstant.Finally,aftertheneckinghaspropa- gatedthroughthewholespecimen,thedeformationbecomeshomogeneousagain( > 4 : 4 )and thestresswillbeincreasing.TheaforementionedstagesareshowninFig.6.15. 6.10Conclusion Inthiswork,amicro-mechanicalmodelbasedontheconceptofnetworkdecompositionisdevel- opedtodescribetheconstitutivebehaviorofDNgelsinlargedeformations.Inthismodel,the DNgelisconsideredasanassemblyoftheandsecondnetworks,whichinteractswitheach otheratlargedeformations.Thenetworkandtheinteractionmodearederivedbyadvanc- ingthepreviouslydevelopednetworkevolutionmodelofcarbonblackrubber.Permanent damageisconsideredforthesenetwork,todescribethestresssoftening,whichisaconsequence oftwosimultaneousprocedures,(i)debondingofchains,and(ii)partialdisentanglementofthe secondnetworkchainsfromnetworkfragments.Thefullnetworkmodelisusedforthesec- ondnetwork,toachievehyper-elasticbehavioruptoverylargestretches.Uponintegrationofthe networkmodelsinallspatialdirectionsandsummingthecontributionofthetwonetworks,a3D 121 representationofthepolymermatrixisobtained.Theperformanceofthemodelisillustratedby comparingitsresultswithasetofexperimentaldata,selectedtorevealthesoftening inthematerial.Theresultsoftheproposedmodelshowagoodagreementwiththeexperimental data.Inadditiontothesimplicityoftheproposedmodel,itsperformancemakesitasuitableoption forcommercialandindustrialapplications. 122 APPENDIX 123 Derivationof S n and C n Inthissection,necessaryequationsforevaluating S n and C n aregiven.ThefreeenergyoftheDN gelisexpressedas M = 1 N + 12 N + 2 N = X j =1 ; 12 jN + 2 N = X j =1 ; 12 " k X i =1 ! i " N j d i max Z n max j n min j d i max P ( n ) c 1 n; d i dn ## + N 2 KT k X i =1 ! i c 2 n 2 ; d i (6..1) Wherethesummationisusedtodescribethebehaviorofthenetworks1and12,astheir governingequationsaresimilar,withonlyminordifferencesinnitionofsomeparameter.In ordertocalculate S n and C n ,wehave S n D D =2 @ M @ C = X j =1 ; 12 " 2 k X i =1 ! i " N j d i max Z n max j n min j d i max P ( n ) @ c 1 n; d i @ C dn ## +2 N 2 KT k X i =1 ! i @ c 2 n 2 ; d i @ C (6..2) 124 C n D D D D =4 d 2 M d C 2 = X j =1 ; 12 ( 4 N j k X i =1 ! i " d i max Z n max j n min j d i max P ( n ) @ 2 c 1 n; d i @ C 2 dn + @ d i max @ C d i max d i Z n max j n min j d i max P ( n ) @ c 1 n; d i @ C dn d i max @n min j d i max @ C d i max d i P n min j d i max @ c 1 n min j d i max ; d i @ C #) +4 ˛ p N 2 KT k X i =1 ! i @ 2 c 2 n 2 ; d i @ C 2 (6..3) Thederivativesusedintheaboveequations,couldbecalculatedasfollows @ c 1 ( n; ) @ C = KT 1 2 R ^ D D (6..4) @ c 2 ( n; ) @ C = KT p n 2 2 D D (6..5) @ 2 c 1 ( n; ) @ C 2 = KT 1 2 1 @ ^ @C 1 2 3 ^ !! D D D D (6..6) @ 2 c 2 ( n; ) @ C 2 = KT 1 2 1 @ @C 1 2 3 D D D D (6..7) @ ^ @ C = R 2 2 tn + 1 1+ t 2 n @ @t D D (6..8) @ @ C = R 2 @ @t D D (6..9) 125 @n min 1 ( ) @ C = R 2 ˘ D D (6..10) @n min 12 ( ) @ C = 1 ˘ 2 D D (6..11) @ @ C =( j 1+ @n min ( ) @ C P n min j d i max n min j d i max R n max j n min j d i max P ( n ) ndn D D (6..12) ThermodynamicConsistency Sincethestrainenergyofthegelmatrix M isbyonlyoneinternalvariable,namely d max ,onecanrewrite M as M = M ( C ; max max max )= ~ M ( F ; max max max )= 1 ( C ; max max max )+ 12 ( C ; max max max )+ 2 N ( C ) ; (6..13) where max max max = ˆ d max : d 2 V 3 ^j d j =1 ˙ : (6..14) ThesecondlawofthermodynamicscanbereducedtotheClausius-Duheminequalitytoshowthe thermodynamicconsistencyofthemodelinanarbitrarydirection d @ d max M _ d max ! 0 8 d : (6..15) Themaximumstretchremainsconstantduringunloadingandreloading.Therefore, _ d max =0 in unloading-reloadingwhile _ D max > 0 intheprimaryloading.Thus,satisfactionoftheClausius- 126 Duheminequalityduringtheloadingissuftoprove(6..15),asonecanwrite @ M @ d max 0 8 d (6..16) Withrespectto(6)and(8),equation(6..15)yields @ M @ d max = @ d 1 @ d max + @ d 12 @ d max 0 8 d : (6..17) Withoutlosinggenerality,(6..17)canbeprovedforanarbitrarydirection d ofprimaryloading oforinteractionnetwork.Forthesakeofbriefness, d max and d arereplacedby x inprimary loadingandinordertotakethederivationofsummationsinthemodel,thesummationsarereplaced bytheirequivalentintegration.Using(17),onecanfurtherobtain @ d i @ d max = @ d i @x = v p N i 2 6 4 d i ( x ) dx Z D i n ( x ) c ( n;x ) P i n ( n ) dn dn i min ( x ) dx i ( x ) c ( n i min ( x ) ; x ) P i n ( n i min ( x )) 3 7 5 (6..18) where n 1 min ( x )= 1 ˘ xv 1 3 p r 0 , dn 1 min ( x ) dx = 1 ˘ v 1 3 p r 0 , n 2 min ( x )= 1 ˘ x 2 v 1 3 p , dn 1 min ( x ) dx = 2 ˘ v 1 3 p x and d i ( x ) dx = A i ˘ v 1 3 p P i n ( n i min ( x )) n i min ( x ) i 1+ i ( x ) R D i n ( x ) nP i n ( n ) dn : (6..19) InEq.(6..19), A 1 = r i 0 and A 2 =2 x .Bysubstituting(6..19)in(6..18),onecanobtain 127 @ d i @x = N i A i ˘ v 2 3 p P i n ( n i min ( x )) 2 6 4 n i min ( x ) i 1+ i ( x ) R D i n ( x ) nP i n ( n ) dn Z D i n ( x ) c ( n;x ) P i n ( n ) dn i ( x ) c ( n i min ( x ) ; x ) 3 7 5 0 (6..20) As 1 , N 1 A i ˘ v 2 3 p P i n ( n min ( x )) > 0 ,and x ) > 0 ,(6..20)holdsifonlywehavethefollowing inequality n i min ( x ) R D i n ( x ) nP i n ( n ) dn Z D i n ( x ) c ( n;x ) P i n ( n ) dn c ( n i min ( x ) ; x ) 0 (6..21) (6..21)canberewrittenas n i min ( x ) Z D i n ( x ) c ( n;x ) P i n ( n ) dn c ( n i min ( x ) ; x ) Z D i n ( x ) nP i n ( n ) dn 0 (6..22) As n i min ( x ) and c ( n i min ( x ) ; x ) arenotfunctionsof n ,theycanbemovedinsidethesummation, thus Z D i n ( x ) P i n ( n ) n i min ( x ) c ( n;x ) n c ( n i min ( x ) ;x ) 0 (6..23) As n i min ( x ) n andthestrainenergyoftheshortestchainisalwayshigherthantheenergy oftherestofthechains ( c ( n;x ) ˝ c ( n i min ( x ) ;x )) ),onecanconclude n i min ( x ) c ( n;x ) c ( n i min ( x ) ;x ) n 0 forall n 2 D i n ( x ) .Whilethebracketintheinequality(6..23)isless thanzeroforallchainlengths( n ),theproposedmodelholdstheconditionofthethermodynamic consistency. 128 CHAPTER7 SUMMARYANDFUTUREWORKS Themainobjectivesofthisstudyweretodevelopaconstitutivemodelforcross-linkedelastomers inlargedeformations.Inthischapter,thedissertationissummarizedforeachsection.In thepartofthisresearch,wefocusedonthederivationofatheoryofpolymerphysicswith adjustableaccuracyandcomputationalcost.Inthefollowingpart,thedevelopedtheoriesareused toproposeconstitutivemodelsinamodularplatformbasistopredictthecomplexbehaviorof cross-linkedelastomers. 7.1GeneralRemarks Ł In chapter3 ,newaccurateapproximationfamiliesoftheNon-GaussianPDF,entropicforce, andstrainenergyofasinglechainaresubsequentlydevelopedtodescribethemechanicsof apolymerchain.Todate,mostofthemicro-mechanicalnon-Gaussianconstitutivemodels areoftendevelopedusingtheKGdistributionfunction,whichisderivedfromthestorder approximationofthecomplexRayleigh'sexactFourierintegraldistribution.However,KG functionisshowntobeonlyrelevantforlongchainsandbecomesextremelyinaccuratefor thechainswithlessthan40segments.TheproposedapproximationsofNon-GaussianPDF, strainenergyandentropicforcewithsimilarlevelsofcomplexityareatleast10timesmore accuratethanKGapproximationsandthusareanexcellentalternativeoptiontobeusedin micro-mechanicalconstitutivemodels. Ł In chapter4 ,anovelapproachisdevelopedthatcanprovideafamilyofapproximation 129 functionsforILFwithdifferentdegreesofaccuracy.Asimpleprocedureispresented,which cantakecurrentapproximationfunctionswithanasymptoticbehaviorandenhancethemby additionofapowerseriesoftheirinducederror.Thetotalerroristhuscorrelatedwith numberoftermsinthepowerseries.Wefurtherproposeddifferentapproachestoreduce thetermsofthepowerseriesandincreasetheaccuracy,theproposedapproachisapplied tofourdifferentclassesofILFapproximationsandshowsimprovements.The accuracy/complexitytrade-offforthefamilyofILFapproximationsgeneratedbythepro- posedapproachiscomparedagainstthoseofotherapproachestoshowthesuperiorityofthe proposedmodel.Theleveloferrorinthismethodcanreachtoavalueaslowas0.02%. Ł In chapter5 ,amicro-mechanicalmodelisdevelopedtocharacterizetheconstitutivebe- haviorofDNelastomersinquasi-staticlargedeformations.Thismoduleoftheplatformis focusedondescribingthenon-linearbehaviorandpermanentdamageinelastomers.The mainsourceofthedamageinthematerialisassumedtobeanirreversiblechaindetachment andbreakageofthechainsinthenetwork.Theproposedmodelenablesustodescribe thedamageandthewayitthemicro-structureofthematerial.Themodelisvali- datedwithuni-axialloadingandunloadingexperimentsoftheDNelastomers.Theproposed modelcontainsafewmaterialconstantsandshowsagoodagreementwithcyclicuni-axial testdata. Ł In chapter6 ,amicro-mechanicalmodelbasedontheconceptofmodularplatformisde- velopedtodescribetheconstitutivebehaviorofDNelastomerswithneckinginstability.In thisplatform,themodelisconsideredasanassemblyofthethreedifferentnetworks.The networkandtheinteractionmodearederivedbyadvancingthepreviouslydeveloped networkevolutionmodelofcarbonblackrubber.Thehyper-elasticfullnetworkmodel isutilizedforthesecondnetwork.Thematrixbehaviorisdividedintothreepartsincluding Œ Pre-necking -Thenetworkisdominantintheresponseofthegelatthethisstage. Œ Necking -Thebreakageofthenetworktosmallernetworkfractions(clusters) 130 inducesthestresssofteningobservedatthisstage.Thedisentanglementofthesecond networkchainsfrombrokennetworkchainsisconsideredasthemaincontributor totheresponseofgelattheneckingstage. Œ Hardening -Limitingstretchofthelongchainsinsecondnetworkisthemainreason ofthehardeninginlargedeformations.Thecontributionofclustersdecreasesduring theneckingasthesecondnetworkstartshardening. Thenumericalresultsoftheproposedmodelarevalidatedandcomparedbyuni-axialcyclic tensileexperimentaldataofDNgels.Finally,aimplementationofthepro- posedmodelispresentedtosimulatetheinitiationandpropagationoftheneckinginstability. 7.2PotentialFutureResearch Inlinewiththisstudy,severalnewquestionsoutsidethescopeofthisdissertationmayarise.Some ofthesequestionsthatrequirefurtherinvestigationsandcanbeastartingpointforfuturestudies arelistedbelow: Ł Fatiguelifetimepredicationoftheelastomericmaterialsiswellstudiedinrecentyears,in whichthereportedmechanicalbehaviorduringcyclicloadingshowsadamage accumulationinthematerial.However,thereisstillthegapintheliteratureregardingthe modellingofmaterialstresssofteningduringthecyclictest. Ł Someofnewlydevelopedcross-linkedelastomericmaterialshavetheabilitytohealwithout anyexternalintervention.Thereareafewstudiesthattriedtopredictdamagingandhealing processofthesematerials.However,ourunderstandingofthephysicalnatureofthehealing overtimehasstillremainedinconclusive. Ł Theexperimentalresultsshowthatthematerialbehaviormaydifferfromsampletosample. Currentpracticeintheconstitutivemodelingofthematerialistoconsidertheaveragere- sponseasatargettopredict.However,theuncertaintyintheresponseofthematerialcan 131 causelossinthesensitiveapplications.So,theadditionoftheuncertaintyfeature totheproposedframeworkcanpredictthebehaviorofthematerialwithdifferentlevelsof desireduncertainty. Ł Micro-mechanicalmodelingofthetime-dependentbehaviorofelastomersisstillchalleng- ingduetotheextremenonlinearnatureoftheirresponse.Inordertounderstandthenature ofthistime-dependentbehavior,thebehaviorofasingleisolatedpolymerchaincanbein- vestigated.Severaladvancementsonnumericalmolecularmodellingaremadeinrecent yearswhichcanhelpusunderstandandvalidatethemicro-structuralbehaviorofthema- terial.Thisunderstandingcanbeintegratedbydevelopedframeworkstopredictnonlinear timedependentbehaviorofthematerial. Ł Severalnewelastomericmaterialsaredevelopedinrecentyears,inwhichthebehaviorof thematerialisimprovedbyaddingorinterpenetratingmultiplenetworksinthe matrix.Thedevelopedframeworkcanbeenhancedtobeusedinthedesignofanewmaterial. Byunderstandingtheeffectsofeachcomponentintheresponseofthematerial,onecan optimizethebehaviorofthematerialbasedontherequired 132 BIBLIOGRAPHY 133 BIBLIOGRAPHY [1] R.E.Webber,C.Creton,H.R.Brown,andJ.P.Gong.LargestrainhysteresisandMullins effectoftoughdouble-networkhydrogels. Macromolecules ,40(8):2919,2007. [2] B.C.MarchiandE.M.Arruda.Generalizederror-minimizing,rationalinverselangevin approximations. MathematicsandMechanicsofSolids ,0(0):1081286517754131,2018. [3] T.Nakajima,Y.Fukuda,T.Kurokawa,T.Sakai,U.Chung,andJ.P.Gong.Synthesisand fractureprocessanalysisofdoublenetworkhydrogelswithanetwork. ACSMacroLetters ,2:518Œ521,2013. [4] S.ShamsEs-haghiandR.A.Weiss.Finitestraindamage-elastoplasticityindouble-network hydrogels. Polymer ,103:277Œ287,2016.NewPolymericMaterialsandCharacterization MethodsforWater [5] PierreMillereau,EtienneDucrot,JessM.Clough,MeredithE.Wiseman,HughR.Brown, RintP.Sijbesma,andCostantinoCreton.Mechanicsofelastomericmolecularcomposites. ProceedingsoftheNationalAcademyofSciences ,115(37):9110Œ9115,2018. [6] LAnand.Aconstitutivemodelforcompressibleelastomericsolids. ComputationalMe- chanics ,18(5):339Œ355,1996. [7] MaryCBoyceandEllenMArruda.Constitutivemodelsofrubberelasticity:areview. Rubberchemistryandtechnology ,73(3):504Œ523,2000. [8] ADorfmannandRayWOgden.Aconstitutivemodelforthemullinseffectwithpermanent setinparticle-reinforcedrubber. InternationalJournalofSolidsandStructures ,41(7):1855Œ 1878,2004. [9] JamesEMarkandBurakErman. Rubberlikeelasticity:amolecularprimer .Cambridge UniversityPress,2007. [10] LeonardMullins.Softeningofrubberbydeformation. Rubberchemistryandtechnology , 42(1):339Œ362,1969. [11] L.R.G.Treloar. Thephysicsofrubberelasticity .OxfordUniversityPress,USA,1975. [12] JianyouZhou,LiyingJiang,andRogerEKhayat.Amicro-macroconstitutivemodelfor viscoelasticityofelastomerswithnonlinearviscosity. Journalofthe MechanicsandPhysicsofSolids ,2017. [13] VahidMorovatiandRoozbehDargazany.Improvedapproximationsofnon-gaussianprob- ability,forceandenergyofasinglepolymerchain. PhysicalReviewE ,2019. 134 [14] VahidMorovati,HamidMohammadi,andRoozbehDargazany.Ageneralizedapproachto improveapproximationofinverselangevinfunction.In ASME2018InternationalMechan- icalEngineeringCongressandExposition .AmericanSocietyofMechanicalEngineers, 2018. [15] J.S.BergströmandM.C.Boyce.Constitutivemodelingofthelargestraintime-dependent behaviorofelastomers. JournaloftheMechanicsandPhysicsofSolids ,46(5):931Œ954, 1998.citedBy(since1996)358. [16] H.Zecha. ZurBeschreibungdesviskoelastischenVerhaltensvonElastomerenbei Verzerrungen:Experimente,ModellbildungundSimulationen. PhDthesis,Institutfür Mechanik(Bauwesen),LehrstuhlI,UniversitätStuttgart,2005. [17] L.Mullins.Effectofstretchingonthepropertiesofrubber. RubberChemistryandTechnol- ogy ,21:281,1948. [18] L.MullinsandN.R.Tobin.Theoreticalmodelfortheelasticbehaviorof-reinforced vulcanizedrubbers. RubberChemistryandTechnology ,30:551,1957. [19] L.MullinsandN.R.Tobin.Stresssofteninginrubbervulcanizates.partI.useofastrain factortodescribetheelasticbehaviorof-reinforcedvulcanizedrubber. JournalofAppliedPolymerScience ,9:2993,1965. [20] RodrigoDiaz,JulieDiani,andPierreGilormini.Physicalinterpretationofthemullins softeninginacarbon-blacksbr. Polymer ,55(19):4942Œ4947,2014. [21] R.DargazanyandM.Itskov.Constitutivemodelingofthemullinseffectandcyclicstress softeninginelastomers. PhysicalReviewE ,88(1):012602,2013. [22] JulieDiani,MathiasBrieu,andJMVacherand.Adamagedirectionalconstitutivemodelfor mullinseffectwithpermanentsetandinducedanisotropy. EuropeanJournalofMechanics- A/Solids ,25(3):483Œ496,2006. [23] JulieDiani,BrunoFayolle,andPierreGilormini.Areviewonthemullinseffect. European PolymerJournal ,45(3):601Œ612,2009. [24] GMachado,GChagnon,andDFavier.Inducedanisotropybythemullinseffectin siliconerubber. MechanicsofMaterials ,50:70Œ80,2012. [25] SerdarGöktepeandChristianMiehe.AmicroŒmacroapproachtorubber-likematerials.part iii:Themicro-spheremodelofanisotropicmullins-typedamage. JournaloftheMechanics andPhysicsofSolids ,53(10):2259Œ2283,2005. [26] M.Itskov,E.Haberstroh,A.E.Ehret,andM.C.Voehringer.Experimentalobservation ofthedeformationinducedanisotropyoftheMullinseffectinrubber. KautschukGummi Kunststoffe ,59:93,2006. [27] FBueche.Molecularbasisforthemullinseffect. JournalofAppliedPolymerScience , 4(10):107Œ114,1960. 135 [28] RoozbehDargazanyandMikhailItskov.Anetworkevolutionmodelfortheanisotropic mullinseffectincarbonblackrubbers. InternationalJournalofSolidsandStructures , 46(16):2967Œ2977,2009. [29] V.MorovatiandR.Dargazany.Micro-mechanicalmodelingofthestresssofteningin double-networkhydrogels.In ASME2018InternationalMechanicalEngineeringCongress andExposition ,pagesV009T12A031ŒV009T12A031.AmericanSocietyofMechanical Engineers,2018. [30] E.M.ArrudaandM.C.Boyce.Athree-dimensionalconstitutivemodelforthelargestretch behaviorofrubberelasticmaterials. JournaloftheMechanicsandPhysicsofSolids , 41(2):389Œ412,1993. [31] SanjayGovindjeeandJuanSimo.Amicro-mechanicallybasedcontinuumdamagemodel forcarbonrubbersincorporatingmullins'effect. JournaloftheMechanicsand PhysicsofSolids ,39(1):87Œ112,1991. [32] GerhardA.Holzapfel. NonlinearSolidMechanics:AContinuumApproachforEngineer- ing .Johnweily&Sons,2005. [33] JunuthulaNarasimhaReddy. Anintroductiontocontinuummechanics .Cambridgeuniver- sitypress,2013. [34] B.D.ColemanandW.Noll.Thethermodynamicsofelasticmaterialswithheatconduction andviscosity. ArchiveforRationalMechanicsandAnalysis ,13:167,1963. [35] B.ColemanandM.E.Gurtin.Thermodynamicswithinternalstatevariables. TheJournal ofChemicalPhysics ,47:593,1967. [36] H.M.JamesandE.Guth.Theoryoftheelasticpropertiesofrubber. TheJournalofChemical Physics ,11(10):455Œ481,1943. [37] E.M.ArrudaandM.C.Boyce.Athree-dimensionalconstitutivemodelforthelargestretch behaviorofrubberelasticmaterials. JournaloftheMechanicsandPhysicsofSolids. , 41:389,1993. [38] M.C.BoyceandE.M.Arruda.Constitutivemodelsofrubberelasticity:Areview. Rubber ChemistryandTechnology ,73:504,2000. [39] V.N.KhiêmandM.Itskov.Analyticalnetwork-averagingofthetubemodel:. Journalofthe MechanicsandPhysicsofSolids ,95:254Œ269,2016. [40] W.Kuhn.Überdiegestaltfadenförmigermoleküleinlösungen. Kolloid-Zeitschrift , 68(1):2Œ15,1934. [41] F.T.Wall.Statisticalthermodynamicsofrubber. TheJournalofChemicalPhysics , 10(2):132Œ134,1942. [42] L.R.G.Treloar.Thestatisticallengthoflong-chainmolecules. TransactionsoftheFaraday Society ,42:77Œ82,1946. 136 [43] E.GuthandH.Mark.Zurinnermolekularen,statistik,insbesonderebeikettenmolekiileni. MonatsheftefürChemie/ChemicalMonthly ,65(1):93Œ121,1934. [44] H.Yamakawa. Moderntheoryofpolymersolutions .Harper&Row,1971. [45] W.KuhnandF.Grün.Beziehungenzwischenelastischenkonstantenunddehnungsdoppel- brechunghochelastischerstoffe. Kolloid-Zeitschrift ,101(3):248Œ271,1942. [46] A.aandM.F.Beatty.Constitutiveequationsforamendednon-gaussiannetwork modelsofrubberelasticity. Internationaljournalofengineeringscience ,40(20):2265Œ 2294,2002. [47] L.Khalili,V.Morovati,R.Dargazany,andJ.Lin.Micro-mechanicalmodelingofvisco- elasticbehaviorofelastomerswithrespecttotime-dependentresponseofsinglepolymer chains.In ConstitutiveModelsforRubberX ,pages523Œ528.CRCPress,2017. [48] V.MorovatiandR.Dargazany.Micro-mechanicalmodelingofthestresssofteningin double-networkhydrogels. InternationalJournalofSolidsandStructures ,2019. [49] LordRayleigh.Xxxi.ontheproblemofrandomvibrations,andofrandominone, two,orthreedimensions. TheLondon,Edinburgh,andDublinPhilosophicalMagazineand JournalofScience ,37(220):321Œ347,1919. [50] V.MorovatiandR.Dargazany.Animprovednon-gaussianstatisticaltheoryofrubberelas- ticityforshortchains.In ASME2018InternationalMechanicalEngineeringCongressand Exposition ,pagesV009T12A030ŒV009T12A030.AmericanSocietyofMechanicalEngi- neers,2018. [51] R.DargazanyandM.Itskov.AnetworkevolutionmodelfortheanisotropicMullinseffectin carbonblackrubbers. InternationalJournalofSolidsandStructures ,46:2967,2009. [52] M.C.WangandE.Guth.Statisticaltheoryofnetworksofnon-gaussianexiblechains. The JournalofChemicalPhysics ,20(7):1144Œ1157,1952. [53] P.J.FloryandJ.R.Rehner.Statisticalmechanicsofcross-linkedpolymernetworksi.rub- berlikeelasticity. JournalofChemicalPhysics ,11:512,1943. [54] P.D.WuandE.VanDerGiessen.Onimprovednetworkmodelsforrubberelasticityand theirapplicationstoorientationhardeninginglassypolymers. JournaloftheMechanics andPhysicsofSolids. ,41:427,1993. [55] C.Miehe,S.Göktepe,andF.Lulei.Amicro-macroapproachtorubber-likematerials-part i:thenon-afmicro-spheremodelofrubberelasticity. JournaloftheMechanicsand PhysicsofSolids. ,52:2617,2004. [56] R.L.JerniganandP.J.Flory.Distributionfunctionsforchainmolecules. TheJournalof ChemicalPhysics ,50(10):4185Œ4200,1969. 137 [57] V.Morovati,H.Mohammadi,andR.Dargazany.Ageneralizedapproachtogenerateop- timizedapproximationsoftheinverselangevinfunction. MathematicsandMechanicsof Solids ,page1081286518811876,2018. [58] R.Jedynak.Newfactsconcerningtheapproximationoftheinverselangevinfunction. Jour- nalofNon-NewtonianFluidMechanics ,249(SupplementC):8Œ25,2017. [59] V.Morovati,H.Mohammadi,andR.Dargazany.Ageneralizedapproachtoimproveap- proximationofinverselangevinfunction.In ASME2018InternationalMechanicalEngi- neeringCongressandExposition ,pagesV009T12A029ŒV009T12A029.AmericanSociety ofMechanicalEngineers,2018. [60] S.Chandrasekhar.Stochasticproblemsinphysicsandastronomy. Reviewsofmodern physics ,15(1):1,1943. [61] K.Nagai.Elementaryproblemsontheofalinearhighpolymer. Journalof thePhysicalSocietyofJapan ,13(8):928Œ934,1958. [62] C.Hsiung,K.L.Verosub,andA.Gordus.Energy-distributionfunctionforhotatomspro- ducedbynucleartransformations. TheJournalofChemicalPhysics ,41(6):1595Œ1600, 1964. [63] Y.Mao.Finitechain-lengtheffectsinrubberelasticity. Polymer ,40(5):1167Œ1171,1999. [64] P.J.Flory. Principlesofpolymerchemistry .CornellUniversityPress,1953. [65] CorneliusOHorganandGiuseppeSaccomandi.Amolecular-statisticalbasisforthegent constitutivemodelofrubberelasticity. Journalofelasticity ,68(1):167Œ176,2002. [66] C.O.Horgan.Theremarkablegentconstitutivemodelforhyperelasticmaterials. Interna- tionalJournalofNon-LinearMechanics ,68:9Œ16,2015. [67] M.F.Beatty.Anaverage-stretchfull-networkmodelforrubberelasticity. JournalofElas- ticity ,70:65,2003. [68] R.Dargazany,V.N.Khiêm,andM.Itskov.Ageneralizednetworkdecompositionmodelfor thequasi-staticinelasticbehaviorofelastomers. InternationalJournalofPlasticity , 63:94,2015. [69] V.MorovatiandR.Dargazany.Netv1.0:Aframeworktosimulatepermanentdamagein elastomersunderquasi-staticdeformations. SoftwareX ,0:0,2019. [70] P.R.vonLockette,E.M.Arruda,andY.Wang.Mesoscalemodelingofbimodalelastomer networks:constitutiveandopticaltheoriesandresults. Macromolecules ,35(18):7100Œ7109, 2002. [71] A.L.Andrady,M.A.Llorente,andJ.E.Mark.Modelnetworksofend-linkedpolydimethyl- siloxanechains.vii.networksdesignedtodemonstratenon-gaussianeffectsrelatedtolim- itedchainextensibility. TheJournalofChemicalPhysics ,72(4):2282Œ2290,1980. 138 [72] E.VerronandA.Gros.Anequalforcetheoryfornetworkmodelsofsoftmaterialswith arbitrarymolecularweightdistribution. JournaloftheMechanicsandPhysicsofSolids , 106:176Œ190,2017. [73] G.M.GuslerandY.Cohen.Equilibriumswellingofhighlycross-linkedpolymericresins. Industrial&engineeringchemistryresearch ,33(10):2345Œ2357,1994. [74] K.Dusek,A.Choukourov,M.Duskkova-Smrckova,andH.Biederman.Constrained swellingofpolymernetworks:characterizationofvapor-depositedcross-linkedpolymer thin Macromolecules ,47(13):4417Œ4427,2014. [75] L.JareckiandA.Ziabicki.Developmentofmolecularorientationandstressinbiaxially deformedpolymers.i.afdeformationinasolidstate. Polymer ,43(8):2549Œ2559,2002. [76] LeszekJareckiandBeataMisztal-Faraj.Non-linearstress-orientationbehaviorofxible chainpolymersunderfastelongationalw. EuropeanPolymerJournal ,95:368Œ381,2017. [77] K.Pearson.Theproblemoftherandomwalk. Nature ,72(1867):342,1905. [78] R.Dargazany. Multi-ScaleConstitutiveModelingofCarbonBlackFilledElastomers .PhD thesis,FacultyofMechanicalEngineering,RWTHAachenUniversity,2011. [79] PDWuandErikvanderGiessen.Onimproved3-dnon-gaussiannetworkmodelsforrubber elasticity. Mechanicsresearchcommunications ,19(5):427Œ433,1992. [80] LRGTreloar.Thephotoelasticpropertiesofshort-chainmolecularnetworks. Transactions oftheFaradaySociety ,50:881Œ896,1954. [81] MikhailItskov,AlexanderEEhret,andRoozbehDargazany.Afull-networkrubberelastic- itymodelbasedonanalyticalintegration. MathematicsandMechanicsofSolids ,15(6):655Œ 671,2010. [82] AlexanderEEhret.Onamolecularstatisticalbasisforogden'smodelofrubberelasticity. JournaloftheMechanicsandPhysicsofSolids ,78:249Œ268,2015. [83] ACohen.Apadeapproximanttotheinverselangevinfunction. Rheologicaacta ,30(3):270Œ 273,1991. [84] MPuso.Mechanisticconstitutivemodelsforrubberelasticityandviscoelasticity.Technical report,LawrenceLivermoreNationalLab.,CA(US),2003. [85] GrantKeady.Thelangevinfunctionandtruncatedexponentialdistributions. arXivpreprint arXiv:1501.02535 ,2015. [86] JörgenStefanBergström. Largestraintime-dependentbehaviorofelastomericmaterials . PhDthesis,MassachusettsInstituteofTechnology,1999. [87] MikhailItskov,RoozbehDargazany,andKarlHörnes.Taylorexpansionoftheinverse functionwithapplicationtothelangevinfunction. MathematicsandMechanicsofSolids , 17(7):693Œ701,2012. 139 [88] EhsanDarabiandMikhailItskov.Asimpleandaccurateapproximationoftheinverse langevinfunction. RheologicaActa ,54(5):455Œ459,2015. [89] awJedynak.Approximationoftheinverselangevinfunctionrevisited. Rheologica Acta ,54(1):29Œ39,2015. [90] MartinKröger.Simple,admissible,andaccurateapproximantsoftheinverselangevinand brillouinfunctions,relevantforstrongpolymerdeformationsandws. JournalofNon- NewtonianFluidMechanics ,223:77Œ87,2015. [91] BenjaminCMarchiandEllenMArruda.Anerror-minimizingapproachtoinverselangevin approximations. RheologicaActa ,54(11-12):887Œ902,2015. [92] AlainNkenfackNguessong,TibiBeda,andFrançoisPeyraut.Anewbasederrorapproach toapproximatetheinverselangevinfunction. RheologicaActa ,53(8):585Œ591,2014. [93] RafayelPetrosyan.Improvedapproximationsforsomepolymerextensionmodels. Rheo- logicaActa ,56(1):21Œ26,2017. [94] HaroldRWarnerJr.Kinetictheoryandrheologyofdilutesuspensionsofextendible dumbbells. Industrial&EngineeringChemistryFundamentals ,11(3):379Œ387,1972. [95] YanYan,MengnanLi,DiYang,QianWang,FuxinLiang,XiaozhongQu,DongQiu,and ZhenzhongYang.Constructionofinjectabledouble-networkhydrogelsforcelldelivery. Biomacromolecules ,18(7):2128Œ2138,2017. [96] SettimioPacelli,PatriziaPaolicelli,MicheleAvitabile,GabrieleVarani,LauraDi-Muzio, StefaniaCesa,JacopoTirillò,CeciliaBartuli,MartinaNardoni,StefaniaPetralito,etal. Designofatunablenanocompositedoublenetworkhydrogelbasedongellangumfordrug deliveryapplications. EuropeanPolymerJournal ,104:184Œ193,2018. [97] P.Calvert.Hydrogelsforsoftmachines. AdvancedMaterials ,21:743Œ756,2009. [98] W.Hong,X.Zhao,J.Zhou,andZ.Suo.Atheoryofcoupleddiffusionandlargedeformation inpolymericgels. JournaloftheMechanicsandPhysicsofSolids ,56:1779Œ1793,2008. [99] A.Lucantonio,P.Nardinocchi,andL.Teresi.Transientanalysisofswelling-inducedlarge deformationsinpolymergels. JournaloftheMechanicsandPhysicsofSolids ,61:205Œ218, 2013. [100] HiroyukiKamata,YukiAkagi,YukoKayasuga-Kariya,Ung-ilChung,andTakamasaSakai. finonswellableflhydrogelwithoutmechanicalhysteresis. Science ,343(6173):873Œ875, 2014. [101] MuhammadAbdulHaq,YunlanSu,andDujinWang.Mechanicalpropertiesofpnipam basedhydrogels:Areview. MaterialsScienceandEngineering:C ,70:842Œ855,2017. [102] JianPingGong.Whyaredoublenetworkhydrogelssotough? SoftMatter ,6(12):2583, 2010. 140 [103] Md.A.Haque,T.Kurokawa,andJ.P.Gong.Supertoughdoublenetworkhydrogelsand theirapplicationasbiomaterials. Polymer ,53:1805Œ1822,2012. [104] T.Nakajima,H.Furukawa,Y.Tanaka,T.Kurokawa,Y.Osada,andJ.P.Gong.Truechemi- calstructureofdoublenetworkhydrogels. Macromolecules ,42:2184Œ2189,2009. [105] TasukuNakajima,HitomiSato,YuZhao,ShinyaKawahara,TakayukiKurokawa,Kazuyuki Sugahara,andJianPingGong.Auniversalmolecularstentmethodtotoughenanyhydro- gelsbasedondoublenetworkconcept. AdvancedFunctionalMaterials ,22(21):4426Œ4432, 2012. [106] X.Zhao.Atheoryforlargedeformationanddamageofinterpenetratingpolymernetworks. JournaloftheMechanicsandPhysicsofSolids ,60(2):319,2012. [107] IDKülcü,MItskov,andRDargazany.Amicro-mechanicalmodelfornon-linearinelastic behaviorofdoublenetworkhydrogels. ConstitutiveModelsforRubberIX ,9:311,2015. [108] F.Bueche.Molecularbasisforthemullinseffect. JournalofAppliedPolymerScience , 10:107Œ114,1960. [109] G.HeinrichandM.Kaliske.Theoreticalandnumericalformulationofamolecularbased constitutivetube-modelofrubberelasticity. ComputationalandTheoreticalPolymerSci- ence ,7:227,1997. [110] S.GovindjeeandJ.Simo.Amicro-mechanicallybasedcontinuumdamagemodelforccar- bonrubbersincorporatingmullins'effect. JournaloftheMechanicsandPhysics ofSolids ,39:87Œ112,1991. [111] G.Marckmann,E.Verron,L.Gornet,G.Chagnon,P.Charrier,andP.Fort.Atheoryof networkalterationfortheMullinseffect. JournaloftheMechanicsandPhysicsofSolids. , 50:2011,2002. [112] M.Itskov,A.E.Ehret,andR.Dargazany.Afull-networkrubberelasticitymodelbasedon analyticalintegration. MathematicsandMechanicsofSolids ,15:655,2010. [113] Y.Tanaka.Alocaldamagemodelforanomaloushightoughnessofdouble-networkgels. A lettersJournalExploringtheFrontiersofPhysics(EurophysicsLetters) ,78,2007. [114] X.WangandW.Hong.Pseudo-elasticityofadoublenetworkgel. SoftMatter ,7:8576, 2011. [115] R.W.OgdenandD.G.Roxburgh.Apseudo-elasticmodelfortheMullinseffectin rubber. ProceedingsoftheRoyalSocietyofEdinburgh,Section:A ,455:2861,1999. [116] YinLiu,HongwuZhang,andYonggangZheng.Amicromechanicallybasedconstitutive modelfortheinelasticandswellingbehaviorsindoublenetworkhydrogels. Journalof AppliedMechanics ,83(2):021008,2016. 141 [117] TongqingLu,JikunWang,RuisenYang,andTJWang.Aconstitutivemodelforsoft materialsincorporatingviscoelasticityandmullinseffect. JournalofAppliedMechanics , 84(2):021010,2017. [118] HaibaoLu,XiaodongWang,XiaojuanShi,KaiYu,andYongQingFu.Aphenomenological modelfordynamicresponseofdouble-networkhydrogelcompositeundergoingtransient transition. CompositesPartB:Engineering ,151:148Œ153,2018. [119] L.R.G.Treloar. ThePhysicsofRubberElasticity .OxfordUniversityPress,2005. [120] H.Yamakawa.Moderntheoryofpolymersolutions. JournalofPolymerScience,PartB: PolymerPhysics ,10:74,1971. [121] VahidMorovatiandRoozbehDargazany.Animprovednon-gaussianstatisticaltheoryof rubberelasticityforshortchains. PhysicalReviewE ,2019. [122] J.-Y.Sun,X.Zhao,W.RKIlleperuma,O.Chaudhuri,K.H.Oh,D.J.Mooney,J.J.Vlassak, andZ.Suo.Highlystretchableandtoughhydrogels. Nature ,489(7414):133Œ136,2012. [123] CostantinoCreton.50thanniversaryperspective:networksandgels:softbutdynamicand tough. Macromolecules ,50(21):8297Œ8316,2017. [124] ZhipengGu,KeqingHuang,YanLuo,LaibaoZhang,TairongKuang,ZhouChen,and GuochaoLiao.Doublenetworkhydrogelfortissueengineering. WileyInterdisciplinary Reviews:NanomedicineandNanobiotechnology ,pagee1520,2018. [125] JianPingGong,YoshinoriKatsuyama,TakayukiKurokawa,andYoshihitoOsada.Double- networkhydrogelswithextremelyhighmechanicalstrength. Advancedmaterials , 15(14):1155Œ1158,2003. [126] RoozbehDargazany,VuNgocKhiêm,EmadAPoshtan,andMikhailItskov.Consti- tutivemodelingofstrain-inducedcrystallizationinrubbers. PhysicalReviewE , 89(2):022604,2014. [127] G.MarckmannandE.Verron.Comparisonofhyperelasticmodelsforrubber-likematerials. RubberChemistryandTechnology ,79:835,2006. [128] A.E.Ehret,M.Itskov,andH.Schmid.Numericalintegrationonthesphereanditseffecton thematerialsymmetryofconstitutiveequations-acomparativestudy. InternationalJournal forNumericalMethodsinEngineering ,81:189,2010. [129] S.HeoandY.Xu.Constructingfullysymmetriccubatureformulaeforthesphere. Mathe- maticsofComputation ,70:269,2000. [130] EijiKamio,TomokiYasui,YuIida,JianPingGong,andHidetoMatsuyama.In- organic/organicdouble-networkgelscontainingionicliquids. AdvancedMaterials , 29(47):1704118,2017. 142 [131] JianHu,TakayukiKurokawa,TasukuNakajima,TaoLinSun,TiffanySuekama,ZiLiang Wu,SongMiaoLiang,andJianPingGong.Highfractureefyandstressconcen- trationphenomenonformicrogel-reinforcedhydrogelsbasedondouble-networkprinciple. Macromolecules ,45(23):9445Œ9451,2012. [132] Jeong-YunSun,XuanheZhao,WidushaRKIlleperuma,OvijitChaudhuri,KyuHwanOh, DavidJMooney,JoostJVlassak,andZhigangSuo.Highlystretchableandtoughhydrogels. Nature ,489(7414):133,2012. [133] K.HaraguchiandT.Takehisa.Nanocompositehydrogels:Auniqueorganic-inorganicnet- workstructurewithextraordinarymechanical,opticalandswelling/de-swellingproperties. AdvancedMaterials(Weinheim,Germany) ,14:1120Œ1124,2002. [134] I.RikuandK.Mimura.Computationalcharacterizationofmicro-tomacroscopicdefor- mationbehaviorofdoublenetworkhydrogel. KeyEngineeringMaterials ,525:193Œ196, 2013. [135] T.Nakajima,T.Kurokawa,S.Ahmed,W.Wu,andJ.P.Gong.Characterizationofinter- nalfractureprocessofdoublenetworkhydrogelsunderuniaxialelongation. SoftMatter , 9:1955Š-1966,2013. [136] Yang-HoNa,YoshimiTanaka,YasunoriKawauchi,HidemitsuFurukawa,Takashi Sumiyoshi,JianPingGong,andYoshihitoOsada.Neckingphenomenonofdouble-network gels. Macromolecules ,39(14):4641Œ4645,2006. [137] VahidMorovatiandRoozbehDargazany.Netv1.0:Aframeworktosimulatepermanent damageinelastomersunderquasi-staticdeformations. SoftwareX ,10:100229,2019. [138] wJedynak.Acomprehensivestudyofthemathematicalmethodsusedtoapproxi- matetheinverselangevinfunction. MathematicsandMechanicsofSolids ,0(0):0,2018. [139] Thanh-TamMai,TakahiroMatsuda,TasukuNakajima,JianPingGong,andKenjiUrayama. Damagecross-effectandanisotropyintoughdoublenetworkhydrogelsrevealedbybiaxial stretching. Softmatter ,15(18):3719Œ3732,2019. [140] N.TriantafyllidisandEliasC.Aifantis.Agradientapproachtolocalizationofdeformation. I.Hyperelasticmaterials. JournalofElasticity ,16(3):225Œ237,1986. [141] RonHJPeerlings,RenédeBorst,WAMarcelBrekelmans,andJHPDeVree.Gradient enhanceddamageforquasi-brittlematerials. InternationalJournalfornumericalmethods inengineering ,39(19):3391Œ3403,1996. [142] M.G.D.Geers,W.A.M.Brekelmans,andR.Borst.ViscousRegularizationofStrain- LocalisationforDamagingMaterials.In DIANAComputationalMechanics`84 ,pages127Œ 138.SpringerNetherlands,Dordrecht,1994. [143] AnthonyMarais,MatthieuMazière,SamuelForest,AParrot,andPLeDelliou. tionofastrain-agingmodelaccountingforlüdersbehaviorinac-mnsteel. Philosophical Magazine ,92(28-30):3589Œ3617,2012. 143 [144] J.W.HutchinsonandK.W.Neale.Neckpropagation. JournaloftheMechanicsandPhysics ofSolids ,31(5):405Œ426,1983. 144