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Preview On the generalized method of cells and the prediction of effective elastic properties of polymer bonded explosives

On the generalized method of cells and the prediction of effective elastic properties of polymer bonded explosives 2 1 BiswajitBanerjee 1 and Daniel O.Adams 0 2 n a Dept. ofMechanicalEngineering,UniversityofUtah, J 1 50SCentralCampusDrive#2202,SaltLakeCity,UT84112,USA. 1 ] i c s - l r t m . t a m - d n o c [ 1 v 5 3 4 2 . 1 0 2 1 : v i X r a 1Correspondingauthor.E-mail:[email protected] Phone:(801)-585-5239 Fax:(801)-585-9826 Abstract Thepredictionoftheeffectiveelasticpropertiesofpolymerbondedexplosivesusingdirectnumericalsimulationsis computationallyexpensivebecause of the high volume fraction of particlesin these particulate composites(∼0.90) and the strong modulus contrast between the particles and the binder (∼20,000). The generalized method of cells (GMC)isanalternativetodirectnumericalsimulationsforthedeterminationofeffectiveelasticpropertiesofcompos- ites. GMChasbeenshownto bemorecomputationallyefficientthanfiniteelementanalysisbasedapproachesfora rangeofcomposites.Inthisinvestigation,theapplicabilityofGMCtothedeterminationofeffectiveelasticproperties ofpolymerbondedexplosivesisexplored.GMCisshowntogenerateexcellentestimatesofeffectivemoduliforcom- positescontainingsquarearraysofdisksatvolumefractionslessthan0.60andamoduluscontrastofapproximately 100. However,forhighvolumefractionandstrongmoduluscontrastpolymerbondedexplosivessuchasPBX9501, the elastic propertiespredictedbyGMCare foundto beconsiderablylowerthanfinite elementbasedestimatesand experimentaldata. Simulationsofmodelmicrostructuresareperformedtoshowthatnormalstiffnessesareunderes- timatedbyGMCwhenstress-bridgingduetocontactbetweenparticlesisdominant. Additionally,thecomputational efficiency of GMC decreases rapidly with an increase in the number of subcells used to discretize a representative volumeelement.TheresultspresentedinthisworksuggestthatGMCmaynotbesuitableforcalculatingtheeffective elastic propertiesof high volume fractionand strong moduluscontrast particulatecomposites. Finally, a real-space renormalization group approach called the recursive cell method (RCM) is explored as an alternative to GMC and showntoprovideimprovedestimatesoftheeffectivepropertiesofmodelsofpolymerbondedexplosives. Keywords:EffectiveProperties,ParticulateComposites,HighVolumeFraction,StrongModulusContrast,Stress- Bridging,MethodofCells,Real-SpaceRenormalizationGroup 1 Introduction Thegeneralizedmethodofcells(GMC) (Aboudi1996,Paley&Aboudi1992)isasemi-analyticalmethodofdeter- miningtheeffectivepropertiesofcomposites.Inthismethod,arepresentativevolumeelement(RVE)ofthecomposite 1 underconsiderationisdiscretizedintoaregulargridofsubcells. Equilibriumandcompatibilityaresatisfiedonanav- eragebasisacrosssubcellsusingintegralsoversubcellboundaries. GMCgeneratesamatrixofalgebraicexpressions containinginformationaboutsubcellmaterialproperties. Theeffectivestiffnessofthecompositecanbeobtainedby invertingthismatrix. OneadvantageofGMCoverothernumericaltechniquesisthatthefullsetofeffectiveelasticpropertiesofacom- positecanbecalculatedinonestepinsteadofsolvinganumberofboundaryvalueproblemswithdifferentboundary conditions. GMChasalso beenfoundto be morecomputationallyefficientthatfinite elementcalculationsforfiber reinforcedcomposites(Aboudi1996,Wilt1995),sincefarfewerGMCsubcellsthanfiniteelementsarenecessaryto obtainthesamedegreeofaccuracy. Theproblemofdiscretizationisalsominimizedsincearegularrectangulargrid isusedinGMC. Thegeneralizedmethodofcellsisdiscussedbrieflyinthiswork.Effectivestiffnessespredictedbythismethodare comparedwithaccuratenumericalpredictionsforsquarearraysofdisks(Greengard&Helsing1998).Themethodis thenappliedtotwo-dimensionalmodelsofageneralpolymerbondedexplosiveandtothemicrostructureofPBX9501 using a two-step proceduresimilar to thatof Low et al. (Low et al. 1994). GMC estimates of elastic propertiesare comparedwithpredictionsfromdetailedfiniteelementcalculations.TheperformanceofGMCisexploredforseveral microstructureswithcontactingparticlesandsomeshortcomingsofthemethodareidentified. Finally,therecursive cellmethod(RCM)isexploredasanalternativetodirectGMCcalculationsinthepredictionofeffectivepropertiesof polymerbondedexplosives. 2 The generalized method of cells Figure 1 shows a schematic of the RVE, the subcells and the notation (Aboudi 1991) used in GMC. In the figure, (X ,X ,X )istheglobalcoordinatesystemoftheRVEand(x(α),x(β),x(γ))isthecoordinatesystemlocaltoasubcell 1 2 3 1 2 3 denotedby(αβγ). Itisassumedthatthedisplacementfunctionu(αβγ)varieslinearlywithinasubcell(αβγ)andcan i bewrittenintheform u(αβγ)(x(α),x(β),x(γ))=w(αβγ)(X ,X ,X )+Φ(αβγ)x(α)+Θ(αβγ)x(β)+Ψ(αβγ)x(γ) (1) i 1 2 3 i 1 2 3 i 1 i 2 i 3 2 where i represents the coordinate direction and takes the values 1, 2 or 3; w(αβγ) is the mean displacement at the i centerofthesubcell(αβγ);andΦ(αβγ),Θ(αβγ),andΨ(αβγ)areconstantslocaltothesubcellthatrepresentgradients i i i ofdisplacementacrossthesubcell. Thestrain-displacementrelationsforthesubcellaregivenby 1 ǫ(αβγ) = (∂ u(αβγ)+∂ u(αβγ)) (2) ij 2 i j j i where∂ = ∂/∂x(α),∂ = ∂/∂x(β),and∂ = ∂/∂x(γ). Sincepolymerbondedexplosivesareisotropicparticulate 1 1 2 2 3 3 composites,thefollowingbriefdescriptionofGMCassumesthattheRVEiscubicandallsubcellsareofequalsize. If eachsubcell(αβγ)hasthesamedimensions(2h,2h,2h)thentheaveragestraininthesubcellisdefinedasavolume averageofthestrainfieldoverthesubcellas 1 h h h ǫ(αβγ) = ǫ(αβγ)dx(α)dx(β)dx(γ) (3) D ij ES 8h3 Z−hZ−hZ−h ij 1 2 3 Theaveragestraininthesubcellcanbeobtainedintermsofthedisplacementfieldvariables. Itisassumedthatthere iscontinuityoftractionattheinterfaceoftwosubcells. Thedisplacementsandtractionsareassumedtobeperiodic at the boundariesthe RVE. Applying the displacementcontinuity equationson an average basis over the interfaces betweensubcells,theaveragestrainintheRVEcanbeexpressedintermsofthesubcellstrains. Theaveragesubcell stressescanbeobtainedfromthesubcellstrainsusingthetractioncontinuityconditionandthestress-strainrelations ofthematerialsinthesubcells. ArelationshipbetweenthesubcellstressesandtheaveragestrainsintheRVEisthus obtained. Fororthotropic,transverselyisotropicorisotropicmaterials,theapproachdiscussedaboveleadstothedecoupling ofthenormalandshearresponseoftheRVE.Thisdecouplingleadstotwosystemsofequationsrelatingthesubcell stresses and the average strains in the RVE. For the normal components of strain, the system of equations can be 3 writtenas M1 M1 M1 T H 0 0  1 2 3 1       M2 M2 M2 T =2N0hǫ i +2NHhǫ i +2N0hǫ i (4)  1 2 3 2   11 V   22 V   33 V                   M3 M3 M3 T  0 0 H  1 2 3 3                whereN isthenumberofsubcellspersideoftheRVE.Thecorrespondingsystemofequationsfortheshearcompo- nentsisoftheform M4 0 0 T H 0 0   12        0 M5 0 T =2N20hǫ i +2N2Hhǫ i +2N20hǫ i (5)   23   12 V   23 V   13 V                    0 0 M6T  0 0 H   13                In equations (4) and (5) the M matrices contain material compliance terms. The T matrices contain the average subcellstresses. ThevectorHcontainsthedimensionsofthesubcells.Thus,asparsesystemofequationsofsize3N2 isproducedthatrelatesthesubcellstressestotheaveragestrainsintheRVE.Afterinvertingtheseequationsandwith some algebraicmanipulation,explicitalgebraicexpressionsfor the individualtermsofthe effectivestiffnessmatrix canbeobtained. Thesestress-strainequationsthatrelatetheaverageRVE stressestotheaverageRVEstrainsareof theform hσ i Ceff Ceff Ceff 0 0 0 hǫ i  11 V  11 12 13  11 V  hσ22iV C1ef2f C2ef2f C2ef3f 0 0 0  hǫ22iV            hσ33iV C1ef3f C2ef3f C3ef3f 0 0 0  hǫ33iV   =   (6)      hσ23iV  0 0 0 C4ef4f 0 0 2hǫ23iV           hσ31iV  0 0 0 0 C5ef5f 0 2hǫ31iV           hσ12iV  0 0 0 0 0 C6ef6f2hǫ12iV      where Ceff are the terms of the effective stiffness matrix. Details of the algebraic expressions for these terms have ij beenpublishedbyotherresearchers(Pindera&Bednarcyk1997). InGMC,thenumberofequationstobesolvedequalsthenumberofsubcellsraisedtothedthpower,wheredis 4 thenumberofdimensionsintheproblem. Asaresult,thecomputationalefficiencyofGMCdecreasesasthenumber ofsubcellsincreases. Thisissuehasbeenpartiallyresolved(Orozco1997)byidentifyingthesparsitycharacteristics of the system of equations and by using the Harwell-Boeing suite of sparse solvers. The computational efficiency of GMC has been further improvedafter a reformulation(Pindera & Bednarcyk1997, Bednarcyk& Pindera 1997) that takes advantage of the continuity of tractions across subcells to obtain a system of O(N2) equations in three dimensions. Due to decoupling of the normal and shear response of the RVE, the shear components of the stiffness matrix obtainedfromGMCaretheharmonicmeansofthesubcellshearstiffnessesandoftheform N N N 1/Ceff =1/N3 1/C(αβγ). (7) 66 66 XXX α=1β=1γ=1 BednarcykandArnold(2001)suggestthatthislackofcouplingmakesforan“ultra-efficient”micromechanicsmodel. However,thislackofcouplingcanleadtogrossunderestimationofshearmoduliforhighvolumefractionandhigh moduluscontrastmaterialssuchaspolymerbondedexplosives.Recently,researchers(Williams&Aboudi1999,Gan et al. 2000)have attempted to solve the problemby using higher order expansionsfor the displacementand by ex- plicitlysatisfyingbothsubcellequilibriumandcompatibility. However,theseapproachesdecreasethecomputational efficiencyofGMCconsiderablyandarenotexploredinthiswork. 3 Validation - square arrays of disks Inthissection, estimatesofeffectivepropertiesfromGMCarecomparedwithaccuratenumericalresultsforsquare arraysofdisks. SquareRVEscontainingsquarearraysofdisksexhibitsquaresymmetry. Thetwo-dimensionallinear elasticstress-strainrelationfortheseRVEscanbewrittenas hσ i K +µ(1) K −µ(1) 0 hǫ i  11 V  eff eff eff eff  11 V  hσ22iV=Keff−µ(ef1f) Keff+µ(ef1f) 0  hǫ22iV  (8)           hσ12iV  0 0 µ(ef2f)2hǫ12iV      5 whereK isthetwo-dimensionaleffectivebulkmodulus,µ(1) istheeffectiveshearmoduluswhenashearstressis eff eff appliedalongthediagonalsoftheRVE,andµ(2)istheeffectiveshearmoduluswhenashearstressisappliedalongthe eff edgesoftheRVE.Thesethreeeffectivemodulihavebeendeterminedaccurately,usinganintegralequationapproach, byGreengardandHelsing(1998)forsquarearraysofdiskscontainingarangeofdiskvolumefractions. TocomparetheeffectivemodulipredictedbyGMCwiththosefromtheintegralequationcalculations(Greengard &Helsing1998),RVEscontainingdiskvolumefractionsfrom0.10to0.70werecreated.TheseRVEswerediscretized into64×64equalsizedsubcells. TheeffectivestiffnessmatrixofeachRVEwascalculatedusingGMC.Finally,the two-dimensionaleffectivemoduliforeachRVEwerecalculatedfromtheeffectivestiffnessmatrixusingtherelations K =0.5(Ceff+Ceff), µ(1) =0.5(Ceff−Ceff), µ(2) =Ceff. (9) eff 11 12 eff 11 12 eff 66 Figure 2 shows the moduli predicted by GMC and those from the integral equation method of Greengard and Helsing(G&H)fordiskvolumefractionsfrom0.10to0.70. Thematerialpropertiesofthedisksandthebinderused inthecalculationsareshowninTable1. Theeffectivebulkmoduli(K )anddiagonalshearmoduli(µ(1))obtained eff eff fromtheGMCcalculationsarewithin4%ofthoseobtainedbytheintegralequationmethodforallvolumefractions up to 0.60. At a volume fraction of 0.70, the GMC predictions for bulk modulus and diagonal shear modulus are 4% and 11% less, respectively. For the shear modulusµ(2), the GMC predictions are around 4% to 10% less than eff theestimatesofGreengardandHelsingforvolumefractionsfrom0.10to 0.60. Thedifferenceisaround24%fora volumefractionof0.70. TheseresultsshowthatGMCestimatesarequiteaccurateforcompositescontainingsquarearraysofdiskswith volumefractionsupto0.60,confirmingresultsreportedelsewhere(Aboudi1996). Inthenextsection,GMCisused todeterminetheeffectivepropertiesofmodelsofpolymerbondedexplosivesandtheresultsarecomparedtodetailed finiteelementcalculationsandexperimentaldata. 6 4 Modeling polymer bonded explosives Polymer bonded explosive (PBX) materials typically contain around 90% by volume of particles surrounded by a binder. The particles consist of a mixture of coarse and fine grains with the finer grains forming a filler between coarsergrains.ModelingthemicrostructureofthesematerialsisdifficultduetothecomplexshapesofHMXparticles andthelargerangeofparticlesizes. Two-dimensionalapproximationsofthemicrostructureofPBXsbasedondigital images(Benson & Conley1999)have beenused to studysome aspects of the micromechanicsof PBXs. However, suchmicrostructuresaredifficulttogenerateandrequirecompleximageprocessingtechniquesandexcellentimage qualitytoaccuratelycapturedetailsofthematerial.AcombinationofMonteCarloandmoleculardynamicstechniques havealsobeenusedtogeneratethree-dimensionalmodelsofPBXs(Baer2001). Microstructurescontainingspheres and oriented cubes have been generated using these techniques and appear to represent PBX microstructures well. However,thegenerationofmicrostructuresusingdynamics-basedmethodsisextremelytimeconsumingwhentight particlepackingisrequired,asisthecaseforvolumefractionsabove0.70. Comparisonsoffiniteelementpredictionswithexactrelationsfortheeffectivepropertiesofcomposites(Banerjee 2002)haveshownthatdetailedfiniteelementestimatescanbeusedasabenchmarktochecktheaccuracyofpredic- tionsfromGMC.Inthisinvestigation,manuallygeneratedPBXmicrostructurescontainingsymmetricallydistributed circularparticlesareusedinitiallytocompareGMCandfiniteelementpredictions. Thetwo-dimensionalmicrostruc- turescontain90%particlesbyvolumeand use two particle lengthscales. Two-dimensionalmodelscontainingran- domlydistributedcircularparticlesthatreflecttheactualparticlesizedistributionofPBX9501arenextmodeledwith GMCandtheresultscomparedtofiniteelementestimates. Thematerialpropertiesusedfortheparticles,thebinder,andPBX9501inthesecalculationsareshowninTable2. ThesepropertiescorrespondtothoseofHMX(theexplosiveparticles),thebinder,andPBX9501at25oCandastrain rateof0.05/s(Wetzel1999). 7 4.1 Simplified models ofPBX materials GMCandfiniteelementcalculationswereperformedforthesix,manuallygenerated,simplifiedmodelmicrostructures ofpolymerbondedexplosivesshowninFigure3. Theserepresentativevolumeelements(RVEs)containoneorafew relativelylargeparticlessurroundedbysmaller particlesto reflectcommonparticlesize distributionsof PBXs. The volumefractionofparticlesineachofthesemodelsisaround0.90±0.005.Thebindermaterialsurroundsallparticles inthesixmicrostructures. FortheGMCcalculations,asquaregridwasoverlaidontheRVEstogeneratesubcells. Twodifferentapproaches were used to assign materials to subcells before the determination of effective properties of the RVE. In the first approach,referredto as the “binarysubcellapproach”,a subcellwas assignedthe materialpropertiesofparticlesif morethan50%ofthesubcellwasoccupiedbyparticles.Binderpropertieswereassignedotherwise.Figure4(a)shows aschematicofthebinarysubcellapproach.Inthesecondapproach,calledthe“effectivesubcellapproach”,amethod ofcellscalculation(Aboudi1991)wasusedtodeterminetheeffectivepropertiesofasubcellbasedonthecumulative volumefractionofparticlesinthesubcell(Banerjeeetal.2000).Figure4(b)showsaschematicoftheeffectivesubcell approach.Afterthesubcellswereassignedmaterialproperties,theGMCtechniquewasusedtocomputetheeffective propertiesoftheRVE. Notethattheparticlesarenotresolvedwellwhenmaterialsareassignedtosubcellsinthismannerifthenumber of subcells is small. However, the large size of the matrix to be inverted in GMC limits the number of subcells thatcanbeusedto discretizetheRVE. Ifthebinarysubcellapproachisusedtoassignsubcellmaterials, contacting particlesarecreatedwheretherearenoneintheactualmicrostructure,leadingtothepredictionofhigherthanactual stiffnessvalues.Theeffectivesubcellapproachimprovesuponthebinarysubcellapproachby“smearing”thematerial propertiesattheboundariesofparticlesandthusreducingtheparticlecontactartifactscausedbydiscretizationerrors. ForvalidatingtheGMCresults,detailedfiniteelement(FEM)calculationswereperformedusingsix-nodedtrian- gularelementstoaccuratelymodelthegeometryoftheparticles.Around65,000nodeswereusedtodiscretizeeachof themodels.ThevolumeaveragestressandstrainineachRVEwasdeterminedforappliednormalandsheardisplace- ments. Periodicitywas enforcedthroughdisplacementboundaryconditions. Since these finite elementcalculations servetovalidatetheGMCcalculations,furthermeshrefinementwasexploredandtheresultswerefoundtoconverge 8 thosefromthe65,000nodefiniteelementmodels. Table 3 lists the effective stiffnesses of the six RVEs shown in Figure 3 from GMC and FEM calculations. On average,theGMCcalculationsusingthebinarysubcellapproachpredictvaluesofCeff thatarearound2.5timesthe 11 FEMbasedvalues. ThevaluesofCeff fromtheeffectivesubcellapproachbasedGMCcalculationsareclosertothe 11 FEMestimatesthanthebinarysubcellbasedGMCestimates. TheGMCandFEMestimatesofCeff arequiteclose. 12 ThevaluesofCeff fromGMCareonly10%oftheFEM values. Thelow valuesofCeff areobtainedbecauseGMC 66 66 predicteffectiveshearstiffnessesthatareharmonicmeansofsubcellshearstiffnesses. Models3and4(Figure3)pro- duceagreementinCeff andCeff betweenthebinarysubcellapproachbasedGMCcalculationsandFEM(within5%) 11 12 whereastheotherfourmodelsproduceconsiderabledifferencesinpredictions. Theselargedifferencesareproduced bydiscretizationerrorsintroducedintheGMCapproachthatleadtocontinuousstress-bridgingpathsacrosstheRVEs andhencetoincreasedstiffness. However, if the predicted effective stiffnesses shown in Table 3 are compared with the experimental effective stiffnessofPBX 9501(showninTable2),itcanbeobservedthatthemodelspredictvaluesofCeff andCeff thatare 11 12 around10%oftheexperimentalvalues. Hence,thesesimplifiedmodelsarenotappropriateforthemodelingofPBX 9501.ThenextsectionexploresmodelsbasedontheactualparticlesizedistributionofPBX9501. 4.2 Models ofPBX 9501 CoarseandfineparticlesofHMXareblendedinaratioof1:3byweightandcompactedintheprocessofmanufacturing PBX 9501. Figure 5(a) shows four RVEs of PBX 9501 based on the particle size distribution of the dry blend of HMX (Wetzel 1999) prior to compaction. Figure 5(b) shows four RVEs based on the particle size distribution of pressed PBX 9501 (Skidmore et al. 1998). The larger particles are broken up in the pressing process leading to a larger proportion of smaller particles in pressed PBX 9501. The models of the dry blend have been labeled “DB” whilethoseofpressedPBX9501havebeenlabeled“PP”. GMCcalculationswereperformedonthePBX9501RVEsafterdiscretizingeachRVEinto100×100subcellsand assigningmaterialstosubcellsusingtheeffectivesubcellapproach. InordertovalidatetheGMCpredictions,FEM calculationswerealsoperformedontheRVEsafterdiscretizingeachRVEinto350×350four-nodedsquareelements. 9

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