Modeling of CNT based composites: Numerical IssuesN. Chandra and C. Shet FAMU-FSUCollegeofEngineering,FloridaStateUniversity,Tallahassee,FL32310AMMLObjectiveTodevelopananalyticalmodelthatcanpredictthemechanicalpropertiesofshort-fibercompositeswithimperfectinterfaces.TostudytheeffectofinterfacebondstrengthoncriticalbondlengthlcTostudytheeffectofbondstrengthonmechanicalpropertiesofcomposites.ApproachTomodeltheinterfaceascohesivezones,whichfacilitatestointroducearangeofinterfacepropertiesvaryingfromzerobindingtoperfectbindingAMMLFig.Shearlagmodelforalignedshortfibercomposites.(a)representativeshortfiber(b)unitcellforanalysis(a)(b)ShearLagModel*Prelude1ThegoverningDEWhosesolutionisgivenbyWhereDisadvantages TheinterfacestiffnessisdependentonYoungsmodulusofmatrixandfiber,henceitmaynotrepresentexactinterfaceproperty.kremainsinvariantwithdeformationCannotmodelimperfectinterfaces*OriginalmodeldevelopedbyCox1andKelly21Cox,H.L.,J.Appl.Phys.1952;Vol.3:p.722Kelly, A., Strong Soilids, 2nd Ed., Oxford UniversityPress, 1973,Chap.5.AMMLPrelude2CohesiveZoneModelCZMisrepresentedbytraction-displacementjumpcurvestomodeltheseparatingsurfacesAdvantagesCZM can create new surfaces. Maintains continuity conditions mathematically, despite the physical separation. CZM represents physics of the fracture process at the atomic scale.Eliminates singularity of stress and limits it to the cohesive strength of the the material.It is an ideal framework to model strength, stiffness and failure in an integrated manner.AMMLModifiedShearlagModelThegoverningDEIftheinterfacebetweenfiberandmatrixisrepresentedbycohesivezone,thenEvaluatingconstantsbyusingboundaryconditions,stressesinfiberisgivenbyAMMLComparisonbetweenOriginalandModifiedShearLagModelVariationofstress-strainresponseintheelasticlimitwithrespecttoparameterbTheparameterb definedbydefinestheinterfacestrengthintwomodelsthroughvariablek.InoriginalmodelInmodifiedmodelinterfacestiffnessisgivenbyslopeoftraction-displacementcurvegivenbyInoriginalmodelkisinvariantwithloadinganditcannotbevariedInmodifiedmodelkcanbevariedtorepresentarangeofvaluesfromperfecttozerobondingAMMLComparisonwithExperimentalResultTheaveragestressinfiberandmatrixfaraappliedstraineisgivenbyThenbyruleofmixturethestressincompositescanbeobtainedasFig.Atypicaltraction-displacementcurveusedforinterfacebetweenSiCfiberand6061-AlmatrixForSiC-6061-T6-AlcompositeinterfaceismodeledbyCZMmodelgivenbyWithN=5,andk0=1,k1=10,k2=-36,k3=72,k4=-59,k5=12.Taking smax=1.8 sy,wheresyisyieldstressofmatrixanddmax=0.06dcAMMLFig. Comparison of experimental 1 stress-strain curve for Sic/6061-T6-Alcompositewithstress-straincurvespredictedfromoriginalshearlagmodelandCZMbasedShearlagmodel.1Dunn,M.L.andLedbetter,H.,Elastic-plasticbehavioroftexturedshort-fibercomposites,Actamater.1997;45(8):3327-3340Theconstitutivebehaviorof6061-T6Almatrix21canberepresentedbyComparison(contd.)yieldstress =250MPa,andhardeningparametersh =173MPa,n=0.46.Youngsmodulusofmatrixis76.4GPa.YoungsmodulusofSiCfiberisEf of423GPaResultcomparisonExperimental1Youngsmodulusis105GPaandfailurestrengthisaround515MPaEc115104.41540522(GPa)FailureStrength(MPa)VariableOriginalModifiedAMMLFEAModelThe CNT is modeled as a hollow tube with a length of 200 , outer radius of 6.98 and thickness of 0.4 . CNT modeled using 1596 axi-symmetric elements. Matrix modeled using 11379 axi-symmetric elements.Interface modeled using 399 4 node axisymmetric CZ elements with zero thicknessComparisonwithNumericalResultsFig.(a)Finiteelementmeshofaquarterportionofunitmodel(b)aenlargedportionofthemeshnearthecurvedcapofCNTAMMLLongitudinalStressinfiberatdifferentstrainlevelInterfacestrength=5000MPaInterfacestrength=50MPaAMMLShearStressinfiberatdifferentstrainlevelInterfacestrength=5000MPaInterfacestrength=50MPaAMMLCriticalBondLengthl/2Table1.Criticalbondlengthsforshortfibersoflength200andfordifferentinterfacestrengthsandinterfacedisplacementparameterdmax1value0.15.AMMLCriticalbondlengthvarieswithinterfaceproperty(Cohesivezoneparameters(smax,dmax1)Whentheexternaldiameterofasolidfiberisthesameasthatofahollowfiber,then,foranygivenlengththeloadcarriedbysolidfiberismorethanthatofhollowfiber.Thus,itrequiresalongercriticalbondlengthtotransfertheloadAthigherdmax1thelongitudinalfiberstresswhenthematrixbeginstoyieldislower,hencecriticalbondlengthreducesForsolidcylindricalfibers,atlowinterfacestrengthof50MPa,whenthefiberlengthis600andabove,thecriticalbondlengthoneachendofthefiberexceedssemi-fiberlengthforsomevaluesdmax1tendingthefiberineffectiveintransferringtheloadinterfacestrengthis5000MPaVariationofCriticalBondLengthwithinterfacepropertyinterfacestrengthis50MPaAMMLTable:VariationofYoungsmodulusofthecompositewithmatrixyoungsmodulus,volumefractionandinterfacestrengthEffectofinterfacestrengthonstiffnessofCompositesYoungsModulus(stiffness)ofthecompositenotonlyincreaseswithmatrixstiffnessandfibervolumefraction,butalsowithinterfacestrengthAMMLEffectofinterfacestrengthonstrengthofCompositesTableYieldstrength(inMPa)ofcompositesfordifferentvolumefractionandinterfacestrengthFibervolumefraction=0.02Fibervolumefraction=0.05Yieldstrength(whenmatrixyields)ofthecompositeincreaseswithfibervolumefraction(andmatrixstiffness)butalsowithinterfacestrengthWithhigherinterfacestrengthhardeningmodulusandpostyieldstrengthincreasesconsiderablyAMMLEffectofinterfacedisplacementparameterdmax1onstrengthandstiffnessFig.Variationofstiffnessofcompositematerialwithinterfacedisplacementparameterdmax1fordifferentinterfacestrengths.Fig.Variationofyieldstrengthofthecompositematerialwithinterfacedisplacementparameterdmax1fordifferentinterfacestrengths.AstheslopeofT-d curvedecreases(withincreaseindmax1),theoverallinterfacepropertyisweakenedandhencethestiffnessandstrengthreduceswithincreasingvaluesofdmax1.Whentheinterfacestrengthis50MPaandfiberlengthissmalltheyoungsmodulusandyieldstrengthofthecompositematerialreachesalimitingvalueofthatofmatrixmaterial.AMMLEffectoflengthofthefiberonstrengthandstiffnessFig.VariationofyieldstrengthofthecompositematerialwithdifferentfiberlengthsanddifferentinterfacestrengthsFig.VariationofYoungsmodulusofthecompositematerialwithdifferentfiberlengthsandfordifferentinterfacestrengthsForagivenvolumefractionthecompositematerialcanattainoptimumvaluesformechanicalpropertiesirrespectiveofinterfacestrength.ForcompositeswithstrongerinterfacetheoptimumpossiblevaluescanbeobtainedwithsmallerfiberlengthWithlowinterfacestrengthlongerfiberlengthsarerequiredtoobtainhighercompositeproperties.DuringprocessingitisdifficulttomaintainlongerCNTfiberstraigth.AMMLConclusion1.Thecriticalbondlengthorineffectivefiberlengthisaffectedbyinterface strength. Lower the interface strength higher is theineffectivelength.2.In addition to volume fraction and matrix stiffness, interfaceproperty, length and diameter of the fiber also affects elasticmodulusofcomposites.3.Stiffness and yield strength of the composite increases withincreaseininterfacestrength.4.Inordertoexploitthesuperiorpropertiesofthefiberindevelopingsuperstrongcomposites,interfacesneedtobeengineeredtohavehigherinterfacestrength.AMML