CRACKMORPHOLOGYEVOLUTIONDUETOPYROLYSISANDCOMBUSTIONIN SOLIDS By YenThiNguyen ADISSERTATION Submittedto MichiganStateUniversity inpartialoftherequirements forthedegreeof MechanicalEngineeringŒDoctorofPhilosophy 2019 ABSTRACT CRACKMORPHOLOGYEVOLUTIONDUETOPYROLYSISANDCOMBUSTIONIN SOLIDS By YenThiNguyen Amodelispresentedforprocessesthatcouplethermaldegradationtocracking,witha focusoncrackformationandpropagationduringpyrolysis.Asthepyrolysisfrontprop- agatesintothesample,acharringlayerisleftbehindwhichcontainsvoids,fracturesand defects.Crackspropagatetoreleasetensilestressesaccumulatedwhenthesampleislos- ingitsmass.Theycanintersecteachother,formingloops,whichareisolatedfragments, unabletoprovidestructuralsupport. ThepyrolysiscrackingprocessissimulatedusingFEM(FiniteElementMethod).The FEMcodeisparallelizedusingMPI(MessagePassingInterface)inordertoaccelerate andcapturethedamageonamesoscale.Variousdimensionlessgroupscharacterizing theproblemaredetermined.Parametergroupsarevariedtoinvestigatetheireffectson themorphologyofthecrackpatterns.Thecrackpatternsobtainedfromthenumerical simulationsareusingimageanalysisalgorithmsandfunctionsthatwerede- velopedandimplementedinMATLAB. Thecrackpatternssharesimilarmorphologicalfeatureswithotherpatternsfoundin natureorinlaboratories,suchasthehierarchyofthecrackingarraysofquenchedplates, thepolygonalmudcracks,thetree-likestructuresofrivernetwork,andleafveins.The expressionofthetree-likeorloop-likebehaviorisdependentuponthechoicesofthepa- rameters.Inparticular,astheratiooftensilestrengthtoYoungsmodulusincreases,the crackbehaviorshiftsfromintersectingtowardbranching.Thebehaviorisalso bypossibleanisotropyinthethermaldiffusivity:behaviorsthatrangefromcracksthat spreadouttocracksthatclustertogether.Furthermore,otherquantities,suchascrack spacing,cracklength,crackpropagationrate,loopdirections,junctionanglesandtheir distributions,crackinitiationtime,aswellastheirdependenceonmaterialproperties, arecomputedaswell,whichprovidesadditionalunderstandingofthegoverningmech- anisms. Copyrightby YENTHINGUYEN 2019 ACKNOWLEDGEMENTS Iwouldliketoexpressmydeepgratitudetowardmytwoacademicadvisors,Professor ThomasPenceandProfessorIndrekWichmanforguidingmethroughyearsofmyPhD study.Withtheirenthusiasm,inspirationandexceptionalpatience,theyspentsomuch timeexplainthingsforme,encouragedmeandgavemeinvaluableinstructionsabout bothresearchandteaching.MyPhDworkisimpossiblewithoutyoursupport. IwouldalsoliketothankProfessorSeungikBaekandProfessorLawrenceDrzalwho servedasmycommitteemembers,givingmeinsightfulcommentsandsuggestions. IamgratefultoProfessorGeoffreyRecktenwaldwhohastaughtmeinvaluableknowl- edgeaboutteachingandalsoprovidedgreatguidanceformetocompletethementored teachingproject. IamthankfultomanyotherProfessorswhohadinstructedmeduringthecourseofmy PhDstudyandTA,especiallyProfessorNeilWright,ProfessorAndreBenard,Professor BrianFeeny,ProfessorAhmedNaguib,ProfessorRicardoMejia-Alvarez. IwishtoacknowledgethesupportoftheMSUCompositeVehicleResearch Center(fundedbyTACOMandtheUSArmy)andtheMSUDepartmentofMechanical Engineeringthatmakethisworkmaterialized. Iamthankfultoallofmyfriendswhohavebeenalwaysbesidesme,lendmetheirhands whenIaminneed.ThankstomyfriendAysewhoisalwayslisteningandencouraging me.EspeciallythankstoAnooshehwhotaughtmetheFEMprogram.Thankstothe VietnamesestudentgroupatMSU. Finally,Iwouldlovetosaymanythankstomyfamilywhoarealwayssupportivetome bothandemotionally.Thankstomylittlethreedaughterswhocheermeup andmakemebecomeastrongerperson.Thankstomanyotherpeoplewhosenamesare notlistedhere. v TABLEOFCONTENTS LISTOFTABLES ....................................... viii LISTOFFIGURES ....................................... ix CHAPTER1INTRODUCTIONANDBACKGROUND ................ 1 1.1CombustionandPyrolysis .............................5 1.2Cracksindryingandcooling ...........................12 1.3Physicsbaseoffractureanddamage .......................16 1.4Modellingofcracks .................................20 CHAPTER2AMATHEMATICALMODELFORTHEPYROLYSISCRACKING PROCESS ................................... 24 2.1Governingequationsingeneralcoordinatesystem ...............24 2.2OneprobleminCartesiancoordinatesystem ..................26 2.3TheRectangularSample ..............................28 CHAPTER3NUMERICALRESULTSFORTHEPROBLEMONARECTAN- GULARDOMAIN ............................. 31 3.1AFiniteVolumescheme ..............................31 3.2FiniteElementimplementation ..........................39 3.3Temperatureanddensity ..........................42 3.4Generalbehaviorofcrackevolutionandmorphology .............48 CHAPTER4MORPHOLOGICALANALYSISANDQUANTIFICATIONOFCRACK MORPHOLOGY ............................... 57 4.1Someimageanalysisconceptsandalgorithms .................59 4.2Morphologicalcharacterizationofnetwork-likepatterns ...........67 4.3ofcrackmorphology .......................76 CHAPTER5DIMENSIONLESSGROUPSANDMATERIALPROPERTIES .... 82 5.1ScalingAnalysis ...................................82 5.2MaterialProperties .................................89 CHAPTER6CASESTUDIES ............................... 105 6.1Theeffectofthecrackingthreshold ........................105 6.2Theeffectofthermaldiffusivities .........................113 6.3Theeffectofheatstrength ...........................118 6.4Theeffectofheatedregionsize ..........................121 6.4.1Generalobservations ............................121 6.4.2Theeffectofvarying s c when l = L ....................123 6.5Theeffectofactivationenergy ...........................127 6.6Theeffectofpre-exponentfactor .........................129 vi 6.7Morphologicaldiagram ...............................129 CHAPTER7CRACKSONACIRCULARDOMAIN .................. 134 7.1Aradialheatingproblem ..............................134 7.2Ananalyticalderivationfortheradialstress ...................138 7.3Numericalresultswithoutcracking ........................142 7.4Numericalresultswithcracking ..........................144 CHAPTER8CONCLUSIONSANDRECOMMENDATIONFORFUTUREWORK 149 APPENDICES ......................................... 153 APPENDIXAANANALYTICALSOLUTIONUNDERSIMPLIFIEDCON- DITIONS. .............................. 154 APPENDIXBNUMERICALRESULTSFORTHECASE s C / E = 0.024. ... 155 APPENDIXCAVERAGECRACKSPACINGONTWOHALVESOFTHE SAMPLE ............................... 159 APPENDIXDCASE s C / E = 0.033USINGDIFFERENTMESHSIZES .... 160 APPENDIXEANXFEMFORMULARFORTHETHERMOELASTICFRAC- TUREPROBLEM .......................... 163 APPENDIXFAVARIATIONALFORMULAFORTHEPHASEFIELD MODELOFFRACTURE ...................... 170 BIBLIOGRAPHY ....................................... 173 vii LISTOFTABLES Table1.1:Trescacriteria ...................................17 Table5.1:VariablesandtheirunitsoftheprobleminSection 2.2 withinitialand boundaryconditions 2.3 . .......................83 Table5.2:Literaturestudyofcellulosicmaterialproperties. ..............91 Table5.3:Propertiesparametersofrubber. ........................95 Table5.4:MaterialvaluesofxGnP. .............................97 Table5.5:MaterialvaluesofxGnPcomposites. ......................98 Table5.6:Materialpropertiesofthermosetsandthermoplastics. ...........99 Table5.7:Representativematerialparameters. ......................104 viii LISTOFFIGURES Figure1.1:Theroadmaptoourstudy. ...........................4 Figure1.2:Parkermodel,showingthepartiallycharredcells,theformationof es,andthevariousformsofheattransfer. ...............9 Figure1.3:Linuxclusterbuiltfromtwopersonallaptops. ...............11 Figure2.1:Boundaryconditions.Uppersurface:heatstressfree.Lateral sides:temperature,stressfree.Lowersurface:insulated,roller. ..30 Figure3.1:ThesubvolumeatnodePanditsneighbornodes. .............33 Figure3.2:Thedimensionlesstemperature q =( T T 0 ) / T 0 at t =50,1000,6000 s fromtoptobottomes.Theleftandrightsidesarecoldwalls whilelowerwallisinsulated.Thesampleisheatedattheuppersur- facebyaheattothecenterregionoverthelength l = 0.1 L 44 Figure3.3:Thedistributionofthedimensionlesstemperature q alongthever- ticalmiddlelineofthesamplewhere x = 0.5atdifferenttimesas indicatedinthelegend,inwhich x = x / L and y = y / L ..........45 Figure3.4:Plotsof r fromlefttoright,toptobottomoftheecorrespondsto t =100,300,1000,1500,2000,3000,4500,6000 s ,respectively.Thechar regiongrowsradiallyfromthecenteroftheheatedregion.Closeto t =6000 s ,charregionreachestheinsulatedlowersurfacebutavoids thecoldsidewalls. ...............................46 Figure3.5:Charfrontandpyrolysisfrontareindicatedbycurvedarcs.Their separation d atonelocationisindicatedbythe ! . ............46 Figure3.6:(a)Thedimensionlessdensity r = r / r 0 atthemid-verticalline( x = 0.5,where x = x / L )atdifferenttimes.Itdecreasesfromuncharred ( r = 1.0)tothecharvalue( r = 0.3)overthedistance d between thecharredanduncharredregions.(b)Theevolutionofthelocations that r = 0.3,0.5,1.0and d .The d vs.timecurvewelltoa squarerootfunction. ..............................47 Figure3.7:Maximumprinciplestress s 1 attimes t =150,1000,3000,5500 s when thesampletensilestrength s c ishigherthan s m .Insuchacasethe sampledoesnotdevelopcracks. .......................48 ix Figure3.8:Maximumprinciplestress s 1 .Plotsfromlefttoright,toptobottom correspondto t =75,300,1000,1500,2000,3000,4500,6000 s ,respec- tively.Attheendofthesecondstage, t =75 s ,thecrackinitiates. ...49 Figure3.9: q attheendofthestage t = 18 s when r = 0.995atthemiddleof theheatedregionontheuppersurface. ...................50 Figure3.10: r attheendofthethirdstage t = 116 s whentheuppersurfacestarts charring. .....................................50 Figure3.11:Maximumprinciplestress s 1 at300 s and1000 s .Initiationsitesin- dicatedintheplotsareatthesampleinteriorwherethelocaldensity gradientconcentrates.Initiationactivitystopsatapproximately400 s . .51 Figure3.12:Plotsof r insamplealongwithcrackingpatternat t =75,300,1000, 1500,2000,3000,4500,6000 s .Crackinginititatesat75 s .At t = 100 s , crackshavealreadydevelopedwhiletheuppersurfacehasnotbeen charredyet.Thepyrolysisfront,followedbythecharfront,isalways behindthelongestcracks.Atalatertime,around4500 s ,thelower surfacestartstopyrolyze;crackspacinginthemiddleregionnear thelowersurfacegetslarger.Thisisalsothelocationwhere d islarge andthedensitygradientissmall.Cracksadvanceindirectionsthat areperpendiculartothepyrolysisfront. ...................52 Figure3.13:Thehierachicalstructureofthecrackpattern.Themaximumprinci- plestress s 1 at3000 s .Thelong(indicatedbyl ! )andtheshort (s ! )branchesalternateeachother. .....................53 Figure3.14:Maximumprinciplestress s 1 at6000 s .Twocrackintersectionsat rightanglearemarkedbytheletterTinred. ................54 Figure3.15:Maximumprinciplestress s 1 at700 s and1200 s .Thecracktipin- dicatedstartssplittingat t = 700 s .At1200 s ,ithassplitintotwo branches. .....................................55 Figure3.16:Maximumprinciplestress s 1 at3800,4000,4500 s .Junctionsformed eitherbythenucleationofachildcrackorbykinkingareindicated withredarrows. .................................56 Figure4.1:Segmentlengthillustrationshowingtheneedforthefactor p 2for pixelsjoinedatedges. .............................61 Figure4.2:Onemethodforthejunctionanglebyusingintersectionpoints ofthecrackpatternwithacircle. .......................61 x Figure4.3:ThesamepixelpatternasFig. 4.2 withadifferentmethodfor thejunctionangleswhichaverageanglesoverdifferentpixels. ......62 Figure4.4:Skeletonizedcrackpatternat6000 s .Eachsegmentiscoloredran- domly.Thenumbersplacedinthemiddleofeachsegmentrepresent thesegmenttotalnumberofpixels. ......................63 Figure4.5:ThesameasFig. 4.4 exceptthatshortbrancheslessthan(left) andeleven(right)pixelsareremoved. ....................63 Figure4.6:ZoomedinsectionofFig. 4.4 showsthevaluesoftheanglesateach junction. .....................................64 Figure4.7:Skeletonizedcrackpatternat6000 s .Numbersdisplayedatthelower rightcorneraretotalisolatedbranches,loops,childbranchesrespectively. 65 Figure4.8:Crackpatternwithloopsat6000 s fromFig. 4.7 .Theareaofeach loopregionisrescaledbythedomainareaanditsvalueisdisplayed attheloopcentroid. ...............................65 Figure4.9:Spuriousintersectionpointsaregroupedtogether. .............67 Figure4.10:Intersectionsoftheflcleanedupflskeletonizedcrackpatternat6000 s withanarcofradius r c / H =0.0317,0.1583,0.2375,0.3167,0.4750, 0.6333,0.7917,0.9500,fromlefttoright,toptobottom,respectively. Rednumbersindicateintersectionpoints. ..................68 Figure4.11:Schematicalgorithmforvertexofapolygon.Eachpairof arrowscorrespondstoalocationofmaximumanglechange,which indicatevertexes. ................................71 Figure4.12:Crackpatternwithloopsat6000 s fromFig. 4.7 .Atthecentroidof eachloopregion,therearetwoarrowsindicatingitsinertialprinciple directions.Arrowlengthsarerescaledbyitsprinciplemomentof inertia:theratiooftwoarrowlengthsistheshapefactor. .........73 Figure4.13:Inscribealoopwithanellipse,asshown. ..................73 Figure4.14:Skeletonizedcrackpatternat6000 s .Cracktipsaremarkedbythe lettereinblueandjunctionsaremarkedbytheletterbinred.When onecrackintersectsanother,itstipisreplacedbyanintersectionpoint. .77 Figure4.15:Theevolutionoftotalnumberofcrackjunctions,cracktipsaswell astheirdifference.Thedifferencetrendsareincreasing,remaining constantanddecreasingfromthethirdstagetothestage. ......78 xi Figure4.16:Probabilitydistributionfunctionofsegmentlengthcorrespondingto crackpatternat6000 s inFig. 4.4 . .......................78 Figure4.17:Probabilitydistributionofjunctionanglescorrespondingtocrackpat- ternat6000 s inFig. 4.4 .............................79 Figure4.18:Probabilitydistributionfunctionofsegmentlengthcorrespondingto Fig. 4.5 whenbranchesarethresholded. ...................80 Figure4.19:ProbabilitydistributionofjunctionanglescorrespondingtoFig. 4.5 whenbranchesarethresholded. ........................80 Figure4.20:Theaveragecrackspacingasafunctionofnormalizeddepthradius r c / l whenthecrackpatternat6000 s isthresholdedusingpixels. ..81 Figure6.1:Maximumprinciplestress s 1 at t = 3000 s correspondingtovari- ousvaluesoftensilestrength s c / E ,whichequal0.024,0.0333,0.0467, 0.0600,0.0667,0.0867,0.1000,0.1133,0.1267,0.1333,0.1600,0.1733 fromlefttoright,toptobottom,respectively. ................106 Figure6.2:Crackingpatternforvariousvaluesoftensilestrength s c at3000 s . As s c getslarger,crackmorphologyshiftsfromloop-liketowardwell developedtree-likeorbranchingbehavior. .................107 Figure6.3:Crackingpatternforvarioustensilestrengthat5000 s .As s c gets larger,crackmorphologyshiftsfromloop-liketowardwelldevel- opedtree-likeorbranchingbehavior. .....................108 Figure6.4:Crackingpatternforvarioustensilestrengthat6000 s .As s c gets larger,crackmorphologyshiftsfromloop-liketowardwelldevel- opedtree-likeorbranchingbehavior. .....................109 Figure6.5:Left:Crackinitiationtimevstensilestrength.Left:Partoftheright ethatcorrespondsto s c / s m < 0.8plottedinalog-logscale. .....110 Figure6.6:Evolutionoftotalcracklengthforvarioustensilestrength. ........110 Figure6.7:NonlinearpartoftheFig. 6.6 . .........................111 Figure6.8:(a)Fig. 6.7 withtimeaxisofeachcurveshiftedbyitsinitiationtime t i . (b) a , b areparametersofthecurve log ( l c / l )= alog ( t t i )+ b inEq.( 6.1 ).(c)slope b ofthelinearpartsvs.tensilestrength s c / s m onaloglogscale.(d)valuesof a and b vs. s c / s m . ..............112 xii Figure6.9:Averagecrackspacing g vsdepth r c / l forvarious s c / E asindicated inthelegend. ..................................112 Figure6.10:Probabilitydistributionofsegmentlengthwhenthecrackimageis notthresholded(left)andthresholded(right).Differentcolorscorre- spondtodifferentvaluesof s c / s m asindicatedinthelegend. ......113 Figure6.11:Probabilitydistributionofjunctionanglewhenthecrackimageisnot thresholded(left)andthresholded(right)forvariousvaluesofthe tensilestrengh s c / s m asindicatedinthelegend. ..............113 Figure6.12:Numberofloopschangeswithtimefordifferentvaluesof s c / s m (left)anditsdependenceon s c / s m at t = 1000,2000,3000,4000,5000, 6000 s (right). ..................................114 Figure6.13:Thedistributionofthedimensionlesstemperature q alongthever- ticalmiddlelineofthesamplewhere x = 0.5atdifferenttimesas indicatedinthelegendforthreecasesof a x .Left: a x = 0.1 a 0 x .Mid- dle: a x = a 0 x .Right: a x = 10 a 0 x . ........................115 Figure6.14:Theeffectsof a x ondensityesaretakenat t = 3000 s .Left e: a x = 0.1 a 0 x , P 3 = 10 P 0 3 .Righte: a x = 10 a 0 x , P 3 = 0.1 P 0 3 . .115 Figure6.15:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolid line.Thelocationofthepyrolysisfrontwhere r = 0.99isindicated bythedashedline.Thepyrolysislength d isindicatedbythedottedline. 116 Figure6.16:Varying P 3 = a y / a x viachanging a x .Lefte: a x = 0.1 a 0 x , P 3 = 10 P 0 3 ,cracksgrowindepth.Righte: a x = 10 a 0 x , P 3 = 0.1 P 0 3 , cracksspreadhorizontally. ...........................116 Figure6.17:Thedistributionofthedimensionlesstemperature q alongthever- ticalmiddlelineofthesamplewhere x = 0.5atdifferenttimesas indicatedinthelegendforthreecasesof a y .Left: a y = 0.1 a 0 y .Mid- dle: a y = a 0 y .Right: a y = 10 a 0 y . ........................116 Figure6.18:Varyingboth P 3 = a y / a x and P 6 = H 2 Aexp ( T a / T 0 ) / a y byvary- ing a y ,Densityat t = 2350 s .Lefte: a y = 0.1 a 0 y leadsto P 3 = 0.1 P 0 3 , P 6 = 10 P 0 6 .Righte: a y = 10 a 0 y leadsto P 3 = 10 P 0 3 , P 6 = 0.1 P 0 6 . ...............................117 xiii Figure6.19:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolid line.Thelocationofthepyrolysisfrontwhere r = 0.99isindicated bythedashedline.Thepyrolysislength d isindicatedbythedottedline. 117 Figure6.20:Varyingboth P 3 = a y / a x and P 6 = H 2 Aexp ( T a / T 0 ) / a y byvary- ing a y , t = 2350 s .Lefte: a y = 0.1 a 0 y leadsto P 3 = 0.1 P 0 3 , P 6 = 10 P 0 6 ,cracksgrowhorizontally.Righte: a y = 10 a 0 y leadsto P 3 = 10 P 0 3 , P 6 = 0.1 P 0 6 ,cracksgrowindepthandtherearefewer cracks. ......................................118 Figure6.21:Thedistributionofthedimensionlesstemperature q alongthever- ticalmiddlelineofthesamplewhere x = 0.5atdifferenttimesas indicatedinthelegendforthreecasesof q 0 .Lefte: q 0 = 0.1 q 0 0 . Middlee: q 0 = q 0 0 .Righte: q 0 = 10 q 0 0 . q scaleslinearlywith q 0 . 119 Figure6.22:Density r at3000 s .Varyingheat q 0 affects P 4 = q 0 H / ( k y T 0 ) . Lefte: q 0 = 0.5 q 0 0 , P 4 = 0.5 P 0 4 .Righte: q 0 = 2 q 0 0 , P 4 = 2 P 0 4 . .......................................119 Figure6.23:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolid line.Thelocationofthepyrolysisfrontwhere r = 0.99isindicated bythedashedline.Thepyrolysislength d isindicatedbythedottedline. 120 Figure6.24:Maximumprinciplestressat3000 s .Varyingheat q 0 affects P 4 = q 0 H / ( k y T 0 ) .Lefte: q 0 = 0.5 q 0 0 , P 4 = 0.5 P 0 4 .Righte: q 0 = 2 q 0 0 , P 4 = 2 P 0 4 . ..............................120 Figure6.25:Theeffectof q 0 oncrackspacing. .......................121 Figure6.26:Maximumprinciplestress(fromSection 3.4 )when q 0 = q 0 0 at6000 s . CrackmorphologyinthiseissimilartothatofFig. 6.24 lower ...121 Figure6.27:Maximumprinciplestress(fromSection 3.4 )when q 0 = q 0 0 at1500 s . CrackmorphologyissimilartothatofFig. 6.24 upper. ..........121 Figure6.28:Effectsofvaryingthesizeofheatedregion l andthus P 2 = l / L on density r .Lefte: P 2 = 0.01, A = 100 A 0 , t = 6000 s .Middle e: P 2 = 1.0, t = 3000 s .Righte: P 2 = 0.1, t = 3000 s ......122 Figure6.29:Effectsofvaryingthesizeofheatedregion l oncrackmorphology. Lefte: P 2 = 0.01, A = 100 A 0 , t = 6000 s .Middlee: P 2 = 1.0, t = 3000 s .Righte: P 2 .1, t = 3000 s . ................122 xiv Figure6.30:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolid line.Thelocationofthepyrolysisfrontwhere r = 0.99isindicated bythedashedline.Thepyrolysislength d isindicatedbythedottedline. 123 Figure6.31:Dimensionlessdensity r = r / r 0 at t = 50,100,500,1000,1200,1350, 3000,6000 s when l = L .OtherparametervaluesaregivenbyEq. ( 3.35 ).Ataround t = 30 s ,theuppersurfacestartstopyrolyzeandat around1350 s ,sodoesthelowersurface. ...................124 Figure6.32:Thecrackpatternsanddistributionofthemaximumprinciplestress s 1 at t = 3000 s for s c / E =0.0240,0.0333,0.0467,0.0600,0.0667,0.0867, 0.1000,0.1733. ..................................125 Figure6.33:Crackspacingismeasuredbyplacingalineintothecrackpattern andcountingintersectionpoints.Inthise,thelineisplacedat depth0.5 H . ...................................125 Figure6.34:Theaveragecrackspacing s asafunctionofdepth r c forvariousval- uesof s c asindicatedinthelegend. ......................126 Figure6.35:Theevolutionoftotalcracklength l c forvariousvaluesof s c asindi- catedinthelegendwhen l = L .Thetotalcracklengthisrescaledby thelength l oftheheatedregion,whichisthesamplewidthinthiscase. 126 Figure6.36:Initiationtime t i ( s ) changeswithtensilestrength s c when l = L ,in which s c isrescaledbythemaximumstressvalueofsamplewithout cracks s m 2 = 0.2 E . ...............................127 Figure6.37:Varyingactivationtemperature T a affects P 5 = T a / T 0 and P 6 = H 2 Ae T a / T 0 / a y .Left: r plots.Right: s 1 plots.Uppere: T a = 0.9 T 0 a leadsto P 5 = 0.9 P 0 5 , P 6 = 23.336 P 0 6 .Lowerure: T a = 1.1 T 0 a leadsto P 5 = 1.1 P 0 5 , P 6 = 0.04285 P 0 6 . ....................128 Figure6.38:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolid line.Thelocationofthepyrolysisfrontwhere r = 0.99isindicated bythedashedline.Thepyrolysislength d isindicatedbythedottedline. 128 Figure6.39:Theeffectof T a oncrackspacing. .......................129 Figure6.40:Varying P 6 = H 2 Ae T a / T 0 / a y byvarying A , t = 3000 s .Left: r plots.Right: s 1 plots.Upperes: A = 10 A 0 . P 6 = 10 P 0 6 . t = 3000 s .Lowerse: A = 0.1 A 0 . P 6 = 0.1 P 0 6 . t = 3000 s . ........130 xv Figure6.41:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolid line.Thelocationofthepyrolysisfrontwhere r = 0.99isindicated bythedashedline.Thepyrolysislength d isindicatedbythedotted line.Whilehighervalueof A acceleratesthepyrolysis, d doesnot changedrasticallywith A . ...........................130 Figure6.42:Theeffectof A oncrackspacing.When A = 0.1 A 0 ,cracksevolveat amuchslowerpaceandonlymaketheirwaytohalfofthesample depth.Thesuddenjumpofcrackspacinginthiscaseat r c / l = 1.5 iscausedbyperiodicdoublingseenatearlystageofcrackevolution. Thecase A = 10 A 0 producessimilarmorphologytotheoriginalcase buttheevolutionhappensatafasterrate. ..................131 Figure6.43:Thecompetitiveof P 3 = a y / a x and P 9 = s c / E onthe morphologyofthecrackpatterns.Thehorizontalaxisisthetensile strength s c rescaledbyitsmaximumvalue s m .Theverticalaxisison thelogarithmscale, log 10 ( a x / a y ) whichis log 10 ( P 3 ) .Closetothe originwhere a y / a x islargeand s c / E issmall,thecrackstendtoform loops.Awayfromtheorigin,theytendtodevelopbranches,forming tree-likepattern. .................................133 Figure7.1:Thedistributionofthedimensionlesstemperature q atdifferenttimes from1 s to600 s inradialheatingcondition. .................144 Figure7.2:Thedistributionofthedimensionlessdensity r atdifferenttimesfrom 1 s to600 s inradialheatingcondition. ....................144 Figure7.3:Thedistributionoftheradialstress s rr atdifferenttimesfrom1 s to 600 s inradialheatingcondition. .......................145 Figure7.4:Thedistributionofthetangentialstress s qq atdifferenttimesfrom1 s to600 s inradialheatingcondition. ......................145 Figure7.5:Themaximumprinciplestress s 1 when s c / E = 0.0833at t =2,10,20, 40,80,100,120,160,180,200,240,280 s . ...................146 Figure7.6:Theevolutionofthemaximumprinciplestress s 1 alongwiththe crackingprocesswhen s c / E = 0.0833 t =2,10,20,40,80,100,120, 160,180,200,240,280 s . ............................147 Figure7.7:Maximumprinciplestress s 1 (top)anddensityld r (bottom)when s c / E =0.05(left)and0.0833(right)at t =270 s ................148 xvi FigureB.1:Maximumprinciplestress s 1 .Plotsfromlefttoright,toptobottom, correspondto t =50,100,300,700,1000,1500,2000,2500,3000,4000, 4500,5000,5700,6000 s ,respectively .....................156 FigureB.2: q attheendofthestage t = 18 s when r = 0.995atthemiddleof theheatedregionontheuppersurface. ...................157 FigureB.3:Plotsof r insamplewithcrackingpatternat t =100,300,1000,2000, 3000,4500,5700,6000 s .At t = 100 s ,crackshavealreadydeveloped whiletheuppersurfacehasnotbeencharredyet. .............158 FigureB.4:Left:Loopbehaviordependson P 9 = s c / E .Right:Crackpattern changewith P 3 = a y / a x . ...........................158 FigureC.1:Averagecrackspacingofthewholesampleandaveragecrackspac- ingonthelefthalfofthesample( x < 0.5)fortwocasesofcracking threshold s c / E = 0.024(upper) s c / E = 0.033(lower). ...........159 FigureD.1:Maximumprincipalstress s 1 / E correspondingto s c / E = 0.033at t =100,1000,3000and5000 s fromtoptobottom,respectively.Mesh size2 h e isusedtoproducethissetofsimulation. ..............160 FigureD.2:Maximumprincipalstress s 1 / E correspondingto s c / E = 0.033at t =100,1000,3000and5000 s ,fromtoptobottom,respectively.Mesh size h e isusedtoproducethissetofsimulation.Notethatthisisthe samecrackpatternasshownbyFig. 3.8 ...................161 FigureD.3:Maximumprincipalstress s 1 / E correspondingto s c / E = 0.033at t =100,1000,3000and5000 s ,fromtoptobottom,respectively.The elementsizeusedtoproducethissetofcracksis h e /2. .......161 FigureD.4:Averagecrackspacingusingthreemeshes2 h e , h e and h e /2for s c / E = 0.033 .......................................162 xvii CHAPTER1 INTRODUCTIONANDBACKGROUND Thisstudyinvestigatestheinteractionsbetweencombustionandthechangeofmorphol- ogyduringburningofsolids.Thestudyhasitsoriginpartlyinmilitaryapplicationssuch ascrackdevelopmentinburningpropellantsandcompositepolymervehicleparts.Itis motivatedbythecrackingpatternonthecharredXGnPresiduesamplesthatweretested usingtheconecalorimeterfacilityintheMSUERCCombustionLaboratory.Indailylife, suchbehaviorisfoundinburnedwoodlogs.Weproposeamodelfortheprocessthat involvescombustion,pyrolysisandcracking,withfocusonthemechanismsofcrackfor- mationandpropagationduringpyrolysis.Numericalsimulationsfollowedby cationofcrackmorphologiesrevealfundamentalprinciplesofthecombustioncracking process. Intheliterature,therearestudiesthataddressdifferentaspectsofthemutualinter- actionsbetweenthesolidandgasphasesduringcombustion.Oneoftheworksof thiskindisParker'smodelforwoodpyrolysis[ 1 ].Assumingthecharringsolidstakes atrapezoidedshapewhencracked,Parkerderivedasystemofgoverningequationsfor energyandmassbalance.TheFireDynamicSimulator(FDS)[ 3 ],adynamicsbased sourcecodedevelopedbyNISTandVTTforthesimulationofedrivenbuoyant, canbeusedtosimulatetheburningofsolidobjects.IntheFDS,solidscanpyrolyzeac- cordingtomultiplesimultaneousorserialArrheniuschemicalkineticsreactions.Then theycandisappearorberemovedfromthecomputationaldomainwhentheirdensity reachesaflburnawayfllimit.TheFDSwasalsousedtoinvestigatethestructuralcollapse ofbuildings,suchasthemodelingofWorldTradeCentereeventin2001[ 4 ].Developed mainlyforchemicallyreacting,theFDShasthermalandsubmodelsthathave beenvalidated.ToenhancethesubmodelpyrolysisoftheFDS,thesourcecodeGpyro (A3DGeneralizedPyrolysismodel)[ 5 ],[ 6 ]wasdevelopedbyC.Lautenberger.Gpyro 1 allowsthesolidstochangeitsshapeduetodensityreduction,byshrinking,whichisthe simplestformofmechanicaldeformation.Otherworksthatconsidersmallscale suchas[ 7 ],[ 8 ],[ 9 ],[ 10 ],couplespreadwithamovingcharring,ormelting,inter- face.Insuchmovingpyrolysisinterfacemodels,thesolidsdivideintotwodistinctzones: acharlayernearthesurfacewheresolidsfullydegradedandavirginzonedeeperinside withinsolid,whichpyrolysis/degradationhasnotyetnced.Furthermore,insuch amodel,thetransitionfromvirginmaterialtocharistakentooccuracrossan imallythinfisheetflorsurface,whichistheso-calledpyrolysisfront.The thinreactionfrontisanidealizationoftheactualprocess.Inotherspreadorpyrol- ysismodels[ 11 ],[ 12 ],thetransitionistakentooccuroveraregionofdimension, namelythepyrolysislengthorthereactionzonethickness.Otherstudiesinvestigatethe relationshipbetweencharformationandspread,suchasinthenumericalsimu- lationofpyrolysisandspreadofapineneedles[ 13 ].Inallofthesespread modelsthereisnoaccountingforcrackformationorsurfacedeformation,twoprocesses thatareasdiscussedaboveforwoodlogs,byroutineempiricalobservation. Certainareasofresearchotherthanspread,[ 14 ]havetheoreticallyinvestigated thepropagationofcracksinconvectiveburningofpropellantusingacohesivezone modelwhichrelatedthegaspressuretothecohesiveforceaheadofthecracktips.In studiesofspallingofconcretestructureine[ 16 ],[ 17 ]theprocessesofwatertransport arecoupledwithheattransferandthethermalexpansionoftheevaporatingvapor,which isbelievedtocauseconcretedamage.Somestudieshaveexaminedcharshrinkageand eformationusingtheprincipleofenergyminimization.Forexample[ 18 ]corre- latesthetotalcracklengthtothesurfacedensityandtheshrinkagegradient.Thestudy [ 19 ]calculatesthesurfaceelasticstrainandthesizeofcharblisters.Inthesestudies, cracksareeitherassumedtoexistpriortotheheatingprocessandremainunchanged, ortheyarealtogetherneglected.Amongthemodelsmentionedabove,theonesthatare consideredinmoregreaterdetailinthisthesisare:Parker'smodelofpyrolysis;FDSpy- 2 rolysisofsolidsandofpyrolysisgases,shrinkagestrainduetomassloss.Each modelitselfisnotadequatetoeithercapturethesolidandgas-phaseinteractions nordotheyincludemechanismsforcrackformationinpyrolysis.Here,aninterdisci- plinarymodelisproposedthattiestogetherdifferentprocessesinthetwophases:heat transfer,pyrolysis,andelasticdeformationandfracture.Fig. 1.1 showstheroadmap oftheoverallprocess,whichisthecombinationofdifferentsubmodelslabeledasA,B, C,D,E,F.Theinterconnectionsbetweenthemarerepresentedbyarrows.Theboldar- rowsrepresentstrongcouplingandthethinonessignifyweakcoupling.Amongthese sixsubmodels,fourbelongtothesolidphase:A(HeattransferinSolid),B(Pyrolysis), C(DeformationandStressAnalysis)andD(MechanicalDamage)whiletheothertwo areforthegasphase:E(MomentumHeatandMassTransferinGasPhase),F(Reacting Gas).Differentcategoriesrelatetoeachotherinthefollowingways:(A)providesthe temperaturedistributionforratesofsubstancedecompositionin(B).Thesechemicalre- actions(B)requireenergy(endothermic)tobreakthebondswhichcausethemtoserveas aheatsinkintheenergyequation(A).Thecombustiblegasesarereleased,whichmaybe ignitedwhentheygaincontactwiththeoxygenambientatasufhightempera- ture.Theseexothermicchemicalreactionsinthegasphaseinvolvehundredsofreactants andtheyreleasethermalenergywhichcansupportthe(F).(E),theequationsthat describetransportationofenergy,speciesandmomentumaresolved,providingthedis- tributionofspeciesconcentrationandtemperatureinthegasphase.Thesolidsubstance cangraduallyloseitsmassviapyrolysis,whichservestobuildupthetensilestress(C), whichcanfracturethesamplewhenacertainstressthresholdlimitisexceeded(D).For retardancy,charformationwithoutcrackingisadesirablefeaturebecauseitcan serveasalowthermalconductivitybarrierbetweentheandthevirginmaterial [ 20 ].Cracks(D)modifythepathsthathotin-depthgasesusetoescapefromthesolid matrixtothefreestream(E).Inparticular,withoutes,hotpyrolysisgasesathigh pressurewillbepushedbackintovirginmaterialsatlowerpressurebypressuregradient 3 force.Theymaycondenseandlater,whentheyarereheated,diffusethroughthehotchar layertothesurfacewheretheycanreactwithatmosphericoxygen[ 11 ].However,the presenceofcrackswillcreateaninstantaneousreleasingpathwayforthosegases,which thenenterthefreestream.Similarly,cracksaugmenttheoxygenpathwayintotheinte- riorregionofthedecomposingsolids.Whencharisexposedtooxygen,itcanundergo anoxidationprocesswhichformscarbondioxideandash.Cracksalsoalterthestress distributioninthesolidsample(C)byconcentratingstressesandmayevenformloops whicharethetheisolatedmaterialfragments. Figure1.1:Theroadmaptoourstudy. 4 1.1CombustionandPyrolysis Thethermalprocessinsidesolidscanbedescribedbyusingthecontinuumheatcon- ductionequation,withtheheatsourceorsinktermsfrompyrolysisreactions(either exothermicorendothermic).Thisequationcanbesolvedsubjecttovariousboundary conditions,whichdescribephysicalprocessessuchasradiationfromoraninci- dentheataswellasconvectionandradiationtotheambientgas.Inamodelthat accountsforapyrolysisfrontthatseparatesthecharredfromtheuncharredregion,heat equationsforthecharredandtheuncharredmatricescanbesolvedseparatelyduetothe assumptionthatthereisatransformationofthermalpropertiesfromtheuncharredto charredsubstances.Oneadditionalconditionisrequiredforthistypeofthermalmodel: continuityoftemperatureattheinterface.Thedistributionoftemperaturecanbeob- tainedusingeithernumericaloranalyticalmethods.Whiletheanalyticalsolutionexists onlyforarestrictedsetofproblems,suchasthosethatpossessselfsimilarsolutionsor Fourierrepresentations,numericalmethodscanbewidelyusedanddonotnecessarily requiretheformaldivisionofregions.Numericalmodelscanrangefromsolvingvery detailedsystemsofPDEstogeneratingintegralsolutionsthatincludesurfacetempera- ture,heatpenetrationandpyrolysisdepth.Inthelatercase,atemperatureprusually linearorquadraticisassumed. Concerningchemicalreaction,allhydrocarbonsubstancesdecomposeatratesthatare acceleratedathightemperature.Somepolymers,likewaxandPMMA,transformintoliq- uids(i.ethemelt)beforeevaporatingintogaseousfuels.Others,suchaswood,rubbers, biowastes,...arebrokendownintocombustiblegases(tars)andsolidresidue(char)via multiplechemicalreactionsinaprocesstermedfipyrolysis.flInpre-moderntimes,pyroly- siswasusedtomakecharcoalsfromwoodlogs.Nowadays,pyrolysisisamajormethod forproducinggasfuels,biogas,coke,activatedcarbon,etcandforreducingindustrial waste.Ithasreceivedmuchattentionduetoitsimportanceinindustrialapplications. Thesimplestpyrolysismodelneglectsallchemicalkineticsbutincludesapyrolysisfront 5 byalimitingtemperature[ 21 ].Thisfront,theoreticallydescribedbyalinein2D orasurfacein3Dmodels,dividessolidsintocharredanduncharredregions.According tothismodels,solidschangeitscomponentoveranthinfront.Morecom- plexmodelsarekineticallycontrolled.Thereactionsrangebetweensingestep reactionmodelstomultiplestepsandreactants.Kineticsparametersforthosemodelsare extractedfromTGA(Thermogravimetricanalysis)experiments.Pyrolyzablematerials, whichhaveacharresiduemorethan5%oftheoriginalweighted,forexamplecellulosic, rubber,DGEBA,PET,PEI,PEEK,arecategorizedascharringsubstances.Whileothers suchasPMMA,POM,PLA,PP,HIPS,arenoncharringones. Inapyrolysismodelof[ 11 ]thatincludesconvectiveandradiativeheatexchangewith ambient,theprocessisdividedintofourdifferentheatingstagesaccordingtothechanges intemperature,massandcharvalues.Inparticular,intheinitialheatingstagethe temperatureisconstantlyrisingbuthasnotreachedavaluewhereitcaninduceapy- rolysischemicalreaction.Inthisstagethemassreleasedisnegligiblysmall.Inthe secondstage,massisproducedalongwithacharfrontthatpropagatesintothein- teriorregion.Inthethirdstage,thetemperatureequilibratestoamaximumlimitvalue whenthereisabalancebetweenheatfromtheandconvectiveheatlosstothe ambient.Themassattainsitspeakvalueinthisstage,thenitgraduallydecreases. Thestageismarkedbyathickcharlayerthatformsneartheendoftheprocess. Amongpyrolyzingpolymers,woodisstudiedextensivelybecauseofitswiderangeof applications.Woodisstructuredfromlongchainhydrocarbonpolymers,whosethepri- marycomponentiscellulose.Theothercomponentsarehemicellulose,lignin,magnan andxylan.Owingtoitspolymericandcompositenature,woodpyrolyzesinmultiple stepsintomultipleproducts.Eachofitscomponentshasitsownbondbreakingten- dency.Generallyspeaking,theproductsfromwoodpyrolysisbelongtooneofthethree categories:char/residue,gas( CO , CO 2 , H 2 O )andtar(othervolatilesthangas,richin 1,6anhydrocompound[ 29 ]).Woodpyrolysismodelsareverynumerous[ 23 ],[ 24 ],[ 25 ], 6 [ 26 ],[ 27 ],[ 28 ],varyingfromamodelsolelybasedonthepyrolysistemperature tomodelswithchemicalreactionsthatcanfurtherbeinto:onestep-singlere- actions;onestep-multiplereactions;multiplestep-multiplereactions.Kineticparameters foreachreactionareattainedbyngTGAdataintoArrheniusequations,whichusually taketheform: ¶r ¶ t = A ( r r c ) n e T a T ,(1.1) wherenistheorderofreaction,Aisthepre-exponentialfactorand T a isanactivation energy.Fortheonestep-singlereactionmodel: virginwood ! volatiles + char ,whichis usedinthisthesis.Thepower ( n = 1 ) producesagoodwithmanyexperimentaldata. With n = 1,Eq.( 1.1 )reducestoEq.( 2.2 )inchapter 1 .Moredetailsaboutthepyrolysis parametersandpyrolysismodelsofwoodarepresentedinSection 5.2 . TheParkermodel[ 1 ]considersthepyrolysisofwoodasaprocessofendothermic chemicalkineticsofitscomponents(water,cellulose,lignin,manan,xylan).Initially, thermalenergyisprovidedbytheincidentheattodesorbthewaterandtodecom- posetheotherfourcomponentsintovolatiles(alsocalledtars)andcharwhichisassumed tohavenoweight,i.e, r c = 0.Thusthemasslossrateforeachspeciestakesthefollowing form: ¶r ¶ t = A r n e T a T .(1.2) Whentotaldensityreachesacriticalvaluewhichisequivalentto r c in( 1.1 ),nomass isproduced.Thevolatilesreleasedareassumedtoresideinthesolidastheydiffuse inward,eventuallyreachingthesolidsurfaceandenteringthegasstream.Whenthe surfacemassishighenoughtoigniteandsupportathepyrolysisprocess becomesselfsustainingbyradiationfromtheathightemperature ( T f = 1200 o C ) andthermalconvectionfromthehot,gas.Heatisremovedfromthesolidsurface byconvectionandbyradiationbythecoldambientairattemperature T ¥ .Thetotalheat tothesurfaceisgivenby 7 q 00 = q 0 q h q r (1.3) inwhich q 0 istheincidentheatand q h and q r aretheconvectiveandradiativeheat betweentheambientandtheTheeffectofdeformationisincludedinthe charshrinkagetermandbymassthroughesincharlayer.Thedepthofthe charlayerisasthelocationwherethesoliddensityis90%ofitsoriginalvalue. Thedependenceofthethermo-physicalandthermo-chemicalpropertiesontemperature andsolidmasswastakenbyParkerfromtheliterature.Themeasurementoftheheat ofcombustionandkineticparameterswasdoneaspartofthatwork.Fornumerical calculation,thesolidwasdividedintoparallelsliceswithmovingboundariessothat nosolidmaterialcrossestheboundaries.Thatboundarymovementischaracterizedby ashrinkagefactorvariablethatisafunctionofthecharvalue.Insummary,Parker's modelincludesbothheattransferandchemicalkineticmodelsinsideactivelydegrading, externallyheatedsolids.TheschematicrepresentationofthemodelisshowninFig. 1.2 . Thepyrolysisofwood,likemostotherpolymers,happensoverathicknesslayer, namelythereactionthicknessregion[ 11 ].However,forsomethermoplasticswithvery highactivationenergy,pyrolysissuddenlyoccurswhenthetemperaturereachesacertain thresholdvalueandthematerialinstantaneouslytransformstogasThe thinreactionzonecanstillbeconsideredagoodapproximationforthese kindofsubstances. Pyropolis,Gpyro,ThermaKinarerecentlydevelopednumericalcodesthatinvestigate aspectsofpyrolysissuchasphasetransitionsandmassdiffusion.ThermaKin[ 30 ]solves thecoupledheatandmasstransferofthereleasedspeciesinthesolidphaseduringpy- rolysis.Heattransfermechanismsinsidethesolidincludeconductive,convectiveand radiativeheattransferhandleuptothirtychemicalreactionsofandsecondorder. ThegoverningequationsinThermaKinaretheconservationofmass,energyandspecies withArrheniusreactionrateforthechemicalreactions.Developedforthesimulationof 8 Figure1.2:Parkermodel,showingthepartiallycharredcells,theformationofes, andthevariousformsofheattransfer. pyrolysisofreinforcedcompositesunderconditionsofstandardetests,Pyropo- lis[ 31 ]capturesthechemicalkineticsofmaterialconstituents,solidheattransferand Darcy'slawforgastransferinsidesolids.Pyropolisextracteditskineticfunctionsand theirparametersfromexperimentaldata,i.e,itisanempiricallybasedmodel. ThepyrolysissubmodelinFDS[ 2 ],[ 3 ],whichisonedimensional,issimilartoTher- maKin,butitcanbecoupledtophasemodels.FDScanextractkineticparameters fromTGAmeasurements.InFDS,thepyrolysisgasesarereleasedintoathatcan supportcombustionwhentheambientoxygenconcentrationissufhighenough tosupportthe Charformationinpyrolysisiscomplicated,howeverlittleisknownaboutitsmecha- nisms.Itisusuallydescribedandmodeledasacompetitivereactionwithactivecellulose inwoodpyrolysisandisrelatedtotheresidencetimeofhotvolatiles.Theformationof charispreferredoverthatofcombustiblegasesandtar,atlowtemperature.Thelayer ofcharovertopthesurfaceofmaterialsservesasaprotectivelayeragainstwhich 9 makessomecharringmaterialseffectivelyeretardant.Producingthischarlayerdur- ingburningisonegoalinthemanufactureeresistantmaterials.Arecently developedmodelofcharcrackingbyLi et.al relatesshrinkagestraintothermalshockin pyrolysis([ 18 ]and[ 19 ])inwhichthethermalshockparameterisby[ 135 ]as f T = k s c ( 1 n ) g t E (1.4) Here g t isthethermalexpansioncoef E istheYoung'smodulus, k isthethermal diffusivity, n isPoisson'sratioand s c isthetensilestrengthofthematerial.Tominimize thermalshock,besidechoosingmaterialsthathavehigh f T ,itisimportanttokeepthe temperaturegradientlowandtochangethetemperatureslowly.Lowtemperaturegra- dientscanbeobtainedifthematerialhassufhighthermaldiffusivity. Baroudi et.al [ 32 ]treatscharshrinkageasathermoelasticbucklingproblemandthey calculatethemorphologyofthecharsurface;Li et.al [ 33 ]correlatethelevelofthechar shrinkagegradienttotheheatandambientpressure. Park et.al inthestudyofwoodpyrolysis[ 34 ],assertthatwoodsplittingiscaused byinternalpressure,nonuniformshrinkage,andstructuralweaknessduetocharring. Thesolidphaseisthencoupledtothegasphaseviaheatandmassatthe interface.Inpyrolysiswithoutcombustionmodels,suchastheParkermodel,theis representedbyaheattermwhenthemassreachesacertaincriticalvalue. Insolidfuelcombustionmodels,thepyrolysisgases(pyrolysatesortars)arereleasedinto the,reactingwithambientoxygenandsupportingthe Ingeneral,atypicalmathematicalformulationofspreadingoversolidfuelshas thefollowingingredients:heattransferinsidethesolid(heatconduction);therate chemicalkineticsinthesolid(Arrheniuslaw);masstransferofspeciesinsidethesolid (conservationofmass,Darcy'slaw);heatattheinterface(convection,radiation); massattheinterface;ratechemicalkinetics;heattransferandspeciestransfer; ellipticalformulationoftheinthegasphase[ 8 ],[ 9 ],[ 10 ],[ 12 ]. 10 TheFDSmodel:TheFDSlibrariescanbeeitherdownloadeddirectlyorcompiledfrom sourcecodeThesourcecode,writteninboth f 90and C ++ programminglanguages, isavailablefreeofcostat https :// github . com / firemodels / fds atfreeofcost.Thenitis compiledbyIntelFortranand C ++ simultaneously,producingtwoseparatelibraries, FDSandSmokeview.Theformerservesthecomputationalpurposesandthelatterserves asagraphictooltovisualizeoutputquantities,suchasgeometryandthedistributionof ,temperatureandsmokeparticles.InthisthesistheFDSlibrarywasrunboth intheserialmodeandintheparallelmodeattheMSUiCERparallelcomputingcenter andattheLinuxclusterbuiltfromtwopersonallaptopcomputers.Eachlaptopisai3 dualcorewithtwoprocessors,yieldingatotaloffourprocessors.MPI(MessagePassing Interface)isusedtocommunicatebetweentheparallelprocessors.Fig. 1.3 showsthe LinuxclusterIbuiltfromtwolaptopstorunFDS. Figure1.3:Linuxclusterbuiltfromtwopersonallaptops. Insummary,innoneofthepreviousstudiesisthereanymodelingofthephysical mechanismsthatoriginatethecrackingprocessesbythermallyinducedstressin 11 theheatedmaterial.Therefore,understandingthemechanismofcrackformationinpy- rolyzingsolidsposestheoreticalandcomputationalchallengessincethereare currentlynodetailedpredictivemodelsforthismechanism.Furthermore,todatethe observationofcrackdevelopmentinlaboratoryexperimentshasnotbeenpossiblebe- causeofthehightemperatureofthesolidsample,andtheevenhighertemperatureof thesurfacewhich,alongwiththeemissionofpyrolysisgasesandsmoke,inhibits diagnosticmeasurements.Inaddition,thedirectnumericalsimulationofcracksinthe contextofsurfacepyrolysishasnotbeencarriedoutduetotheabsenceofacomprehen- sivetheoryaswellasthecomplexitystemmingfromtheappearanceofmultiplecracks. 1.2Cracksindryingandcooling Therehas,beenresearchontherelatedtopicofshrinkagecracks,whichmayoccur duringdrying.Concerningshrinkagecracks,modelshavebeendevelopedthatdescribe howsolidsdeforminresponsetodrivingforcessuchashightemperatureinthermoe- lasticmaterials,moistureinthedryingshrinkageoffoodpastes,porepressureindrying colloidalgels,andspeciesconcentrationsincrystals,see[ 35 ].Similarly,stressescanbe developedbythenonuniformshrinkageofamaterialsubjectedtoexternalheatingand subsequentthermaldegradation(pyrolysis)andtheassociatedinducedmassloss.This hypothesisissupportedbyempiricalobservationand,morerecently,byexperi- mentsthathavesoughtcorrelationsbetweencrackingdepth,heatpenetrationdepth,and pyrolysisdepth,see,e.g.,[ 18 ]. Naturehasprovidedexamplesofshrinkagecracksofwhichasamplingisdiscussed here.Ingeothermalscience:crackcolumnsinbasaltcooling[ 38 ];mudcracksinawetrice orinwetclay,[ 55 ],[ 56 ],[ 57 ],[ 65 ],mudpeeling[ 58 ];frozensolidsunderdiurnalforc- ing[ 59 ],ice[ 61 ];seasonalthermalcontractioncracksinpermafrostonMars[ 60 ];ground cracks[ 62 ];dryingsoils[ 63 ],[ 75 ],saturatedsoil[ 64 ].Inbiology:snakeskincracking patternsduetocellgrowth. 12 Thesetypesofcracksarealsogeneratedinlaboratories:parallelcrackpatternsincold thermalshockceramics[ 66 ],[ 67 ];crackingofcolloidal[ 68 ],indryingsuspensions [ 69 ],opalinsuspensions[ 70 ];ringcracksincolloidalcrystalsofsilicasphereson coverglass[ 71 ];cracksincrystaldecomposition;crackpatternsinpolymercrystallization [ 72 ](rectangle,spiral,branching);splittingoffoodspaghetti[ 73 ],udon[ 74 ];crackin concrete[ 76 ],hardenedportlandcementpastes[ 77 ]andinplasticcement[ 78 ].Inart: crackpatternsonglazeceramicoroldpaintings. Chemicaldecompositioncausesmolarvolumedecrease.Thisisobservedinthefor- mationofcracksincrystalgrowthsubstancessuchascrystallinebariumchloride dihydrate[ 79 ]andpowderedpotassiumcopper(II)chloridedihydrate[ 80 ].Inmodeling, thereductionofthemolecularequilibriumdistanceinquenchingorchemicaldecompo- sitionisrepresentedbyaproportionalitybetweentemperaturedifferenceormolarcon- centrationandvolumetricstrainonthecontinuumscale.Ontheotherhand,cracking ofdryingcolloidalsuspensionsisdrivenbyporepressurethustheoverallelasticstress istakentobethesumofthemechanicalstressandtheporesuction.Itisnotedthat theproblemsofquenchinganddirectionaldryingoffoodaregovernedbythediffusion equations,suchastheheatconductionandthecracklengthisfoundtoscalewiththe squarerootoftime,analogoustoselfsimilarvariablesthatappearingeneraldiffusive processes.Intheproblemsofdryingforcolloidalsolids,theporepressureisrelatedto poresizethroughLaplace'slawandtheevolutionofcracklengthfollowsapowerlaw. Ingeneralpractice,crackinginproductsshouldbeavoidedatallcostsbecauseitdete- rioratesthesamplequality,thereby,malfunctioningthedevice.Thereareexceptions,for examplethecrazepatternsinglazeceramics,whichisadesiredaestheticoutcome. Theseitemslistedbelowdescribesvariousshrinkagecrackingmechanismsandthe associatedfailurecriteriaforseveralkindsofmaterialsfoundinmanyreferencestudies. Soil.Surfacetensioneffectsatair-water-soilcontact(suction). Horizontalstressat cracktipismoretensilethanthetensilestrengthofthesoil.(p266).Rankinecondition, 13 [ 81 ] . Thinofcolloidalsilica.Porepressurein Cracksadvanceiftensilestress exceeds s go ,haltifitfallsbelow s stop ,[ 69 ]. Colloidalincapillarytubes.Tensilestress s 0 arisingfromcontractionofthemedium duetocapillaryporepressureEq.[1],page3. Minimizingthelocalstrainenergy density,byratiobetweenlengthandcellsize,[ 82 ]. soil(siltyclay)ground,excavatedresidualsoilground.Soilsuction(pore airpressure,porewaterpressure). Nethorizontalstressexceedstensilestrengthofsoil, [ 62 ]. Finegrainedsoil.Microscopic:variationinforceactingbetweengrains.Macro- scopic:drying-inducedstrain. Cohesivebondbreaksirreversiblywhennormalforce, tangentialforce,momentumoftwodiscreteelementexceedsanenvelope;Eq.[3],[ 83 ]. Type10Portlandcementwithoxidecomposition.Waterevaporationcausecapillary stresses;C-S-Hgeldrying. Shrinkagetensilestressexceedsavalue,[ 77 ]. Concretetensilestrength.Freeshrinkageofsmallelementsofconcretecauseby decreaseofporehumidityasafunctionofhumidity;Eq.[1]. Tensilestrength,[ 84 ] . Cementitiousmaterials.Capillarypressurecausedbycurvedwatersurfacebetween particles.Plasticorcapillaryshrinkage, [ 78 ] . Plainandreinforcedconcrete.Shrinkagestraindependsonmoisturecontent; chemicalshrinkagewhenvolumeofproductsislessthanreactants. VonMisesstress equaltothetensilestrength,[ 85 ] . Concreteringrestrainedbyasteelring.Stressduetorestrainedanddifferential moisturelossshrinkage. Tensilestressexceedscriticalvalue,[ 86 ] . 14 Udonnoodles(wheatandsaltwater).Moistureloss. Tensilestressexceeds criticalvalue,[ 74 ] . Concrete.Chemicalshrinkage,dryingshrinkage. Rankinefailurecriterion,[ 87 ] . Resincomposite.Shrinkageduetopolymerization. Failureofbondingagents;micro- gapformation,[ 88 ] . Granularsoils.Strainrateequalselasticstrainrateplusmetricchangerate.Non- uniformparticlesizechangegeneratesself-equilibratedforces. Crackinitiateswhen strongesttensilecontactsbegintofail,[ 89 ] . Starchandwatermixture.Dessication. Contractionstressesexceedmaterialstrength,[ 38 ]. Opalmadefromsuspensionpolystyrenesphere.Particlemovementinsolidis suppressedduringwettodrysolidtransitionregime. [ 70 ] . BasalticlavaStressesgeneratedbythermalgradients. Firstfractureappears whenamaximumstressisexceeded.Hexagonalcolumnofcrackisformedbytheenergy minimizationprinciple,[ 90 ] . Basalt.Purelytensileandantiplaneshearduringcrackpropagation. [ 91 ] . Mudofclayparticles.Porepressurecausestensilestress,tensilestressgradient causespeelingcracks. CrackpeelsinmodeIandmodeIIoffracturemechanics,[ 58 ] . Spaghetti.Differentialdryingshrinkage. Cracksaresupposedtobecausedbythestress valuesthatexceedfailurestrength,[ 73 ] . (modeling).Saturatedclaysoil.Moistureloss. ModeIstressintensityfactorexceeds criticalvalue(fracturetoughness),[ 64 ] . (experimentandmodeling). ModeIstressintensityfactorislargerthansoilsfracture toughness.Criticalenergyreleaserateisattained,[ 92 ] . 15 (analytical).Colloidal.Shrinkingcolloidalisrestrainedbyarigidsubstrate. Jinte- gralvalueorenergyreleaserateisexceeded,[ 93 ] . 1.3Physicsbaseoffractureanddamage Theliteratureoffracturehasdevelopeddifferenttheoriesaboutmaterialfailure.Among thesearethephenomenalfailurecriterion;fracturemechanics;cohesivezonemodeland damagemechanics.Accordingtothephenomenologicalfailuretheories,brittleandduc- tilematerialsbehavedifferentlyinfailure.Inbrittlematerials,stretchincreaseslinearly withstressuntilsuddenlyruptureoccurs.Ontheotherhand,ductilematerialsstretch linearlywithappliedloaduntilayieldpointafterwhichstretchincreases withoutincreaseofappliedload.Stressesarethenredistributedasthesampledeforms beforeactualfailureoccurs.Failurecriteriaareintodifferentcategoriesbased onnormalstress,shearstress,maximumprinciplestress,ormaximumenergy.Common phenomenologicalfailurecriteriaandtheircontextsarelistedbelow: theTrescaormaximumshearstress(brittle); thevonMisesormaximumelasticdistortionalenergycriterion(ductile); theMohr-Coulombfailurecriterion(cohesive-frictionalsolids); theDrucker-Prager(pressure-dependentsolids); theWillam-Warnke(concrete); theBresler-Pister(concrete); theHankinson(orthotropicmaterialssuchaswood); theHillyieldcriteria(anisotropicsolids); theTsai-Wufailure(anisotropiccomposites); 16 theJohnsonHolmquistdamagemodel(high-ratedeformationsofisotropicsolids); theHoek-Brown(rockmasses); theCam-Clay(forsoils). Themaximumstresscriteria(Rankine'stheory)forbrittlematerials,alsonamedthe maximumstresstheory,statesthatfailurewilloccurifthemaximumprinciplestress ( s 1 , s 2 )reachesacriticaltensile( s t )orcompressivestress( s c ).Thiscanbeexpressed mathematicallyas s c s 1 , s 2 s t .Themaximumstraincriterion(SaintVenant'sthe- ory),forbrittlematerials,statesthatwhenthemaximumprinciplestrainreachesacritical value,failurewilloccur: j s 1 s c n s 2 j s t j j = 1, j s 2 s c n s 1 j s t j j = 1,where n isthePoisson'sratio. Themaximumshearstress(Trescacriteria),forductilematerials,isbasedontheyield stresswhichcausesslippageoflayersthatareoriented45 0 degreetothenormalstress. TheTrescacriteriaaresummarizedintable 1.1 below. Bothintension s 1 > 0, s 2 > 0 s 1 , s 2 < s t Bothincompression s 1 < 0, s 2 < 0 s 1 , s 2 > s c s 1 intension, s 2 incompression s 1 > 0, s 2 < 0 s 1 s t + s 2 s c < 1 s 1 incompression, s 2 intension s 1 < 0, s 2 > 0 s 1 s c + s 2 s t < 1 Table1.1:Trescacriteria Thereisanothertypeoffailuretheoryfortheyieldingofductilematerials,whichisbased onthedistortionalstrainenergy,forexample,theVonMisescriterion: s 2 1 s 1 s 2 + s 2 2 = s 2 f (1.5) 17 orCoulomb-Mohr'stheory: s 1 s t = 1, s 2 s t = 1, s 1 s t = 1, s 2 s c = 1, s 1 s t s 2 s c =+ 1, ortheyield/failurecriterion[ 37 ],forisotropicmaterial,startingfromthegeneralpolyno- mialexpansionofthestresstensor,whichhastheform L = ds kk + zs 2 kk + h s ij s ij (1.6) where d , z , h arethematerialconstitutiveparameters, s ij isthedeviatoriccom- ponentofthestress s ij : s ij = s ij d ij 3 s kk .Theelasticenergymustbepositive whichrequiresthat d = 0.Thus,oneformof L ,whichequaltoelasticenergy,istakenas U = E 2 b s kk E 2 + 3 2 ( 1 b ) s ij E s ij E ,(1.7) where b = 1 2 n 3 (1.8) and n isPoisson'sratio. Christensenassumesthathomogeneousmaterialdoesnotfailunderhydrostaticpressure butcouldfailunderhydrostatictension.Thisrequires z inEq.( 1.6 )tovanish,andthe failurecriterionaccordingtoChristensenis a s kk k + 3 2 ( 1 + a ) s ij s ij k 2 1,(1.9) where k = j s c j ,whichisthecriticalcompressivestressand a = j s c j s t 1forahomoge- neousmaterial,forwhich: 0 s t j s c j 1(1.10) 18 Theductile-brittlebehaviorisdeterminedbythevaluesof a : a < 1: 1 2 s t j s c j 1 Ductile , a = 1: s t j s c j = 1 2 Transition , a > 1:0 s t j s c j 1 2 Brittle . When a = 0,Eq.( 1.9 )becomes 1 2 s ij s ij k 2 3 whichistheMisescriterion.When a ! ¥ ,an extremebrittlecondition,Eq.( 1.9 )becomes 1 2 s ij s ij ks kk 3 .Thosearethetwoextreme limitsof a . Thereareotherfailuretheories,forexample:thetheorybasedonfracturetoughness, stressintensityfactorandthestrainenergyreleaserateoffracturemechanics;thetheory basedonadamagevariableofdamagemechanics.Fracturemechanicswasdevel- opedbyGrif[ 123 ]tostudycrackpropagationinglasses.Inlinearelasticfractureme- chanics(LEFM),crackssurfacesaretractionfree,cracksgrowwhenthestressintensity factororstrainenergyreleaserateexceedsacertainlimittogeneratenewfreesurface.In thecohesivezoneofmodelfracturemechanics,theseparationoccursoveraregionahead ofthecracktipandthepotentialcracksurfaceisboundbyacohesiveforce. UnlikeLEFM,whichassumesthatfailureonlyoccursatthecracktip,thuscanonly predictcrackpropagation,thecohesivezonemodelanddamagemechanicsareableto predictcrackinitiation.Inthecurrentproblem,thecharringsoliddevelopsvoidsand microcrackswhendegradingbeforedevelopingcracks,thereforethecohesivemodelor damagemechanicsprovideadvantagesandmightbefuturework. TheanalyticalsolutionnearcracktipsforcertainproblemsofLEFMcanbederived usingcomplexvariableanalysis.Itisfoundthatthestressneartipofthecrack s is inverselyproportionaltothesquarerootofthedistancefromcracktip r ,as K / ( 2 p p r ) . Here K isthestressintensityfactorthatdependsonloadingconditions,geometry,ect. Thusthestressisatcracktip(where r = 0).Thecrackpropagateswhen K reaches 19 acriticalvalue, K K c .Usinganenergybalanceapproach,acertainquantityofenergy, whichcanbecalculatedfromtheJintegral,mustbereleasedforcracktopropagate. Thecontinuumdamagemodel(CDM),[ 124 ],[ 125 ],describestheevolutionofdamage alongwithstressesandstrainsbyintroducingthedamagevariablewhichmaybeeither scalarortensorial.Thescalardamagevariable D isasthelimitingratioofthe totalareathatcontaindefects,cracksandthetotalarea,asthelatterapproacheszero.The variable D rangesbetween0and1: D = 0correspondstoundamagedstate, D = 1to totallydamagedstateand0 < D < 1describesthepartiallydamagedstate.Innumerical modeling,therearelocalapproachesforcrackpropagation,suchasthediscretecrack incrementapproachwhichresolvesthenearcracktipIntheseapproaches,cracks extendwhencertainquantitieslikestress,strain,energyaheadcracktipreacha limitingvalue.CrackpropagationinthelocalCDMischaracterizedbythereductionof materialstiffness.Thecrackzoneistakenasthelocusofpointswhere D = 1.However, similartootherlocalapproaches,thelocalCDMtheorypossesdiffornumerical modeling,suchasmeshdependenceofthecrackwidthandtheaccuratecomputationof thecrackgrowthrate. 1.4Modellingofcracks Differenttheoreticalornumericalapproachestocrackingcanaseithercon- tinuummodels,suchas,fracturemechanics,damagemechanics,theFiniteElementMethod (FEM),theExtendedFiniteElementMethod(XFEM),ordiscretemodelsincludingdis- creteelements,springblocks,bundlespringblock,electricalfuses,peridynamicsormolec- ulardynamics.Theycanbealsodividedintolocalmodels(classicalFEM,XFEM)which assumesthatamaterialpointonlyinteractswithotherpointsinacloseneighborhood;or nonlocalmodels(peridynamics,moleculardynamics),[ 94 ]whichhavedistanceinterac- tionswithothermaterialpoints.ComparedtocontinuummethodssuchasFEM,discrete modelsrequirenoremeshingandcanhandleaveryhighnumberofnodessincesolving 20 thelinearalgebraicsystemofequationsisnotrequired.Ontheotherhand,FEMisanu- mericalmethodforsolvingPDEsforgeneralmaterials,varyingfromelastictoviscoelastic toplastic,anduseshigherorderelementsforbetterconvergencetowardpotentiallyexact solutions. ThetraditionalFEMdevelopedbasedonLEFMexplicitlytrackscracksbynodes.It requirescrackmeshalignmentandthuscontinuousmeshrnearthecracktip. Theextendedelementmethod(XFEM)developedbyBelytschkoandBlack[ 97 ],[ 98 ] enrichesnodesintheneighborhoodofasmoothcrackusingdiscontinuousHeaviside stepfunctionsandtheasymptoticcracktipfunctions.XFEMhasdemonstratedmesh independenceandthusremeshingisgenerallynotrequired.Theasymptoticcracktip functionsarederivedfromtheoreticalfracturemechanics,givenas F 1 ( r , q )= p rsin ( q /2 ) F 2 ( r , q )= p rcos ( q /2 ) (1.11) F 3 ( r , q )= p rsin ( q /2 ) sin q F 4 ( r , q )= p rsin ( q /2 ) sin q where ( r , q ) isthelocalpolarcoordinatesystematthecracktip.Itisnotedthatthe functionin( 1.11 )isdiscontinuousacrossthecrackface.Thenumericalsimulationsof dryingcracksintreebarkandtwolayermud[ 110 ]producesrealisticpatternswhichvary withparameterssuchasYoung'smodulus,layerthickness,rateofgrowthandshrinkage, thresholdstress.[ 99 ]implementedtheXFEMforthesimulationofthermoelasticcracks. [ 100 ]presentsageneralstructureofanobject-orientedXFEMcodeandstepsforextend- inganexistinggeneralpurposeFEMcodelikeAbaqusintoaXFEMcodewithsmall Theisappliedfor:extendedvariableordegreeoffreedom pernodes,detectingelementscutbythecracks,enrichedstiffnessmatrices,divisionof 21 cutelementsfornumericalintegration.ThelibraryOpenXFEM ++ writteninC ++ were developedby[ 100 ]andtestedforvarious2Dcrackproblems. Phasemodelhasbecomeincreasingpopularwithmanyapplicationsandnu- mericalcodesforvariousproblem,includingfracture.Inphasemodel,the isrepresentedbyascalarthatvariescontinuouslywithlocations.Thereareopen phasesourcecodessuchasMOOSE,FEniCS,OpenPhase,DUNE,FiPy,MICRESS, PACE3D,(see[ 106 ],[ 107 ],[ 108 ],[ 109 ]).SomestudiesderivedanFEMformulaforthe crackparameterandthedisplacementandimplementedtheminAbaqususing Usersubroutines,suchastheworkof[ 104 ].Aphasemodelforporoelasticmaterial isdevelopedby[ 105 ].MoredetailsaboutXFEMandphasemodeloffractureare presentedattheAppendix,Sections E and F . Theboundaryelementmethod(BEM)isalsoanumericalmethodthatcanbeapplied tomodelcracks.BEMresolvestheunknownontheboundaryinsteadofthewhole domain.OneofthemaindrawbacksofBEMisthatthefundamentalsolutionofthepar- tialdifferentialequationsmustbeknownbeforeasitwillbeusedastheweightfunction. Thespringlatticemodel,proposedbyKawai[ 111 ],discretizestheelasticdomain withnodesconnectedbysprings,[ 95 ],[ 96 ].Eachspringischaracterizedbyaspringcon- stantandabreakingthresholdwhichisusuallyofMohr-Coulombtype.Differenttypes oflatticecanbeusedinthemodel,usuallysquareorhexagonal,whichareeitherstruc- turedandrandomlydistributedoranunstructuredmesh.Inpractice,anunstructured meshliketheVoronoitessellationcanavoidmeshbiasedwhensimulatingcrackswhich isaproblemwhenusingsquareandhexagonallattices.Thedegreeoffreedomineach nodeofthelattice,thedisplacements u and v canbeextendedtoincludeanotherdegree offreedomforrotation.Toderivethespringconstantofeachlatticefromtheelasticpa- rameters,thestoredstrainenergyofthelatticecell E cell = 1 2 N b å j = 1 F j u j isequatedtothatof thecontinuummaterial E con = 1 2 R W s# d W .Here F isthespringforcebetweentwonodes inthelattice, N b isthenumberofbondsineachlatticeand j istheindexofeachspring.To 22 modeltheanisotropy,eachspringconstantisallowedtohavedifferentvaluesdepending onthespringorientation.Therearevariousspringmodels,suchastheKirkwoodmodel forisotropicmaterials,theKeatingmodelformaterialswithnegativePoisson'sratio,and theWozniakflbeambendingflapproachforthespringlatticemodel.Thecriteriaforcrack initiationandpropagationinspringmodelsareusuallysimple,e.g,crackingoccursat locationswherethespringbreakswhencertainthresholdvalueisattained. Discretemodelshavebeenappliedandsuccessfullygeneraterealisticcrackpatterns indryingsolids,suchasthediscreteelementmodelingofdryingandcrackingofsoils [ 112 ].Inthosemodels,thedryingprocessismodeledbychangingthespringnatural lengthfollowingthetimeexponentdecayingrule. Thereareotherdiscretemethods,includingmoleculardynamicsandperidynamics, bothofwhicharenonlocal.Whiletheformerisonthenanoscale,thelattercanbeboth microandmesoscale,inwhichthesizerangesnear1 m m .Similartomoleculardynamics, peridynamicsisformulatedwithoutspatialdifferentials,thusitisabletosimulatecracks whichhavediscontinuitiesacrosstheirfaces.Themesoscalemakesitlesscomputation- allyexpensive,whichisanattractivefeature.PeridynamicshasbeencoupledwithFEM forsolvingthethermalshockproblem,suchas[ 113 ],[ 114 ],[ 115 ],inwhichFEMwasused fortheheattransferequationsandperidynamicswasusedtomodelthethermalcracks. Thebondforceisincorporatedwiththethermalexpansionorcontraction.In[ 113 ],the couplingcoefbetweenthermalexpansionandbondforceisdeterminedbyequat- ingtwostrainenergy,insuchwaythatthespringconstantofthediscretespringmodel isfound.Oterkusetal.[ 116 ]derivedaperidynamicsmodelforthefullycoupledthermo mechanicalproblembasedonconservationofenergyandthefreeenergyfunctionofther- modynamics.Failureoccurswhenbondstretchexceedsacriticalstretchvalueandthat bondisbrokenandcrackappearsspontaneously.Omittingtheeffectsofcracksonheat transferinthesestudiesofthermalshockisacommonpractice.Moreover,itis whenthecrackingdirectionisparalleltotheheatdirection. 23 CHAPTER2 AMATHEMATICALMODELFORTHEPYROLYSISCRACKINGPROCESS 2.1Governingequationsingeneralcoordinatesystem Ourmathematicalmodelincludesheattransferinthesolid,materialbreakdown(py- rolysis)underhightemperature,elasticdeformation,andcrackformationinthesolid material.Here,thegasphaseprovides,throughtheactionofahypotheticalthe externalheatforthesolidphasewhichisthefocusofourstudy.Thetemperature T ( ~ x , t ) inageneralcoordinatesystemisdescribedbytheheatconductionequation ¶ T ¶ t = r . ( a r T ) (2.1) where a isthethermaldiffusivity.Therateofpyrolysisisdescribedbythefollowing singlestepdecompositionreaction ¶r ¶ t = A ( r r c ) e T a / T .(2.2) Here A isthepre-exponentialfactor, T a istheactivationtemperatureand r c isthelower boundofthesoliddensity,orthechardensity.Thestresstensor s isrelatedtothestrain tensor # bythestandardlinearelasticityrelation(Hook'slaw): s = C : # m ,(2.3) orequivalently # m = S : s ,(2.4) wherethesubscript m standsforthemechanicalcomponent,andCandSarethefourth orderstiffnessandcompliancetensors,respectively.Thesolidshrinksasitlosesmass 24 duringpyrolysis.Theshrinkageismodeledbytakingtheshrinkagestrain # v asbeing proportionaltotheamountofmassloss, # v = g ( r r 0 ) / r 0 I ,(2.5) where g isthecouplingcoefbetweenmasslossandvolumetricshrinkage.The overallstrain # isthesumofthemechanicalstrain # m causedbystressesandtheshrinkage strain # v , # = # m + # v .(2.6) Moreover,thetotalstrainisrelatedtodisplacementsbythesmalldeformationrelation, whichisgivenby # = 1 2 ( r ~ u +( r ~ u ) T ) ,(2.7) inwhich ~ u isthedisplacementvector.Stressesobeythequasi-steadystateequilibrium equationwhichneglectstheinertialterm: r s = 0.(2.8) Itisassumedthatcracksnucleateandgrowwheneverthemaximumprincipalstress s p reachesathresholdvalue s c ,whichistakenhereasamaterialconstant.Thuscracking occursatlocationswhere s p s c .(2.9) InitialconditionsarerequiredforthetemperatureanddensityTheboundary conditionsforthethermalandthestressproblemcanbeeitherprescribedtemperaturesor displacements(Dirichlet),prescribedheatortraction(Neumann)ormixed(Robin) conditions,suchasrollersupportsorconvectionatboundaries(especiallytheonesbor- deringthegasphase). 25 2.2OneprobleminCartesiancoordinatesystem Considerthemainprobleminarectangulardomain B = f ( x , y ) j 0 < x < L ,0 < y < H g ,whichisenclosedbyaboundary ¶ B = ¶ B 1 S ¶ B 2 S ¶ B 3 S ¶ B 4 ,Inparticular, ¶ B 1 = f ( x , y ) j x = 0,0 y H g , ¶ B 2 = f ( x , y ) j y = 0,0 x L g , ¶ B 3 = f ( x , y ) j x = L ,0 y H g , ¶ B 4 = f ( x , y ) j y = H ,0 x L g . . TheheatconductionEq.( 2.1 )inthiscoordinatesystemcanbewrittenas: ¶ T ¶ t = a x ¶ 2 T ¶ x 2 + a y ¶ 2 T ¶ y 2 ,(2.10) where a x and a y arethethermaldiffusivitiesinthe x and y directions,respectively,which aretakentobeconstants.TheequationforthedecompositionprocessisgivenbyEq. ( 2.2 ). Becauseofthetwo-dimensionalnatureoftheproblemunderconsideration,thestress andstraintensorsaretakentobeoftheform: # = 2 6 6 6 6 4 # xx # xx 0 # xy # yy 0 00 # zz 3 7 7 7 7 5 , s = 2 6 6 6 6 4 s xx s xx 0 s xy s yy 0 00 s zz 3 7 7 7 7 5 .(2.11) Assuminganisotropicandhomogeneousmaterial,thestiffnesstensor C isand onlydependentontwoparamters.ThesearetheYoung'smodulus, E andthePoisson's ratio, n .TheoverallstrainfollowsfromEq.( 2.3 ),Eq.( 2.6 )andEq.( 2.5 ): 26 # xx = 1 E s xx n s yy + s zz + g r r 0 r 0 ,(2.12) # yy = 1 E s yy n ( s xx + s zz ) + g r r 0 r 0 ,(2.13) # zz = 1 E s zz n s xx + s yy + g r r 0 r 0 ,(2.14) # xy = 1 + n E s xy .(2.15) Onemayconsidertwoseparatecasesforanalysis.Thecaseistheconditionofplane straininwhich # zz = 0.Thesecondcaseistheconditionofplanestressinwhich s zz = 0. Forplanestrain,itfollowsfromthecondition # zz = 0andEq.( 2.14 )that s zz = n s xx + s yy E g r r 0 r 0 ,(2.16) andthus s xx , s yy arerelatedto # xx , # yy via: s xx = E 1 + n # xx + E n ( 1 + n )( 1 2 n ) # xx + # yy E ( 1 2 n ) g r r 0 r 0 ,(2.17) s yy = E 1 + n # yy + E n ( 1 + n )( 1 2 n ) # xx + # yy E ( 1 2 n ) g r r 0 r 0 .(2.18) Fortheplanestresscondition,itfollowsfromputting s zz = 0inEq.( 2.12 )to( 2.14 )that s xx = E 1 + n # xx + E n 1 n 2 # xx + # yy E ( 1 n ) g r r 0 r 0 ,(2.19) and s yy = E 1 + n # yy + E n 1 n 2 # xx + # yy E ( 1 n ) g r r 0 r 0 .(2.20) Forbothplanestressandplanestrain,itfollowsimmediatelyfromEq.( 2.15 )that: 27 s xy = E 1 + n # xy .(2.21) Thedisplacementsinthehorizontalandverticaldirectionsare u and v ,respectively, ~ u = ( u , v ) .Thestrainsinthe ( x , y ) planearerelatedto u and v bythestandardrelations, followingEq.( 2.7 ): # xx = ¶ u ¶ x # yy = ¶ v ¶ y (2.22) # xy = 1 2 ¶ u ¶ y + ¶ v ¶ x Thestressequationsofequilibriumforboththeplanestrainandplanestressproblems followEq.( 2.8 ),whichgive ¶s xx ¶ x + ¶s xy ¶ y = 0,(2.23) and ¶s xy ¶ x + ¶s yy ¶ y = 0.(2.24) 2.3TheRectangularSample Toillustratethismodel,considerthefollowingsetofinitialandboundaryconditionsas anexample.Initially,thetemperatureisuniformatthevalue T 0 : t = 0: T = T 0 .(2.25) Thethermalboundaryconditionsaretakentobethefollowing:insulatedatthelower surface ¶ B 4 ;uppersurface ¶ B 2 issubjectedtoaheat q ( x ) ;thetwolateralsides ¶ B 1 and ¶ B 3 aremaintainedattheinitialtemperature T 0 .Theseconditionsreadas 28 ¶ B 1 : T = T 0 ,(2.26) ¶ B 2 : ¶ T ¶ y = 0,(2.27) ¶ B 3 : T = T 0 ,(2.28) ¶ B 4 : k y ¶ T ¶ y = q ( x ) .(2.29) InEq.( 2.29 ),theheatfunctionisasanonzeroconstantoverthecentral regionoftheuppersurface,viz, q ( x )= 8 > > < > > : 0if j x L 2 j > l 2 q 0 if j x L 2 j l 2 . (2.30) Here k y isthethermalconductivityintheydirectionand q 0 istheconstantheatFor thepyrolysisproblem(Eq.( 2.2 )),oneinitialconditionisneeded.Initiallythematerial istakentohaveuniformdensity r 0 .Theboundaryconditionsforthestressproblemare takentobeofthesecondandthethirdtype: ¶ B 1 : s xx = 0, s xy = 0, ¶ B 2 : s yy = 0, s xy = 0, ¶ B 3 : s xx = 0, s xy = 0,(2.31) ¶ B 4 : v = 0, s xy = 0, whichareequivalenttotherollerconditiononthelowersurface B 4 andthetractionfree conditionsontheothersurfaces B 1 , B 2 , B 3 .Theboundaryconditionsdiscussedabove fortheproblemsareillustratedinFig. 2.1 . Fortheplaneproblem,twooftheprincipalstressesoccurinthe ( x , y ) plane 29 Figure2.1:Boundaryconditions.Uppersurface:heatstressfree.Lateralsides: temperature,stressfree.Lowersurface:insulated,roller. s 1 = s xx + s yy 2 + s s xx s yy 2 2 + s 2 xy ,(2.32) s 2 = s xx + s yy 2 s s xx s yy 2 2 + s 2 xy .(2.33) Theremainingprincipalstress,whichisnonzeroonlyintheplanestraincase,isthestress normaltothe ( x , y ) plane s 3 = s zz .(2.34) Themaximumprincipalstressisthen: s p = max ( s 1 , s 2 , s 3 )= max ( s 1 , s 3 ) . 30 CHAPTER3 NUMERICALRESULTSFORTHEPROBLEMONARECTANGULARDOMAIN 3.1AFiniteVolumescheme Thepurposeofthissectionistodevelopthenumericalstencilsthatdiscretizethe governingequations,Eqs( 2.23 ),( 2.24 ),inwhichtherelationsofstressestostrainsand strainstodisplacementsaregivenbyEqs.( 2.22 ),( 2.19 ),( 2.20 ),( 2.21 ),whichsatisfytheset ofboundaryconditionsinSection 2.3 .Thecomputationaldomain B ismeshedusinga structuredgrid.Traditionally,Taylor'struncationcanbeutilizedtoderivesuchstencils, however,withmaterialdepletion,adifferentmethodmustbeused.Following[ 118 ],the numericalstencilsarederivedbyrequiringdirectsatisfactionoftheequationswithina subvolumeinanintegralsense.Thisisrecognizedasaweakerconditionoftheoriginal partialdifferentequations.Let B = N S k = 1 w k ,inwhich w k , ( k = 1, N ) isthesubvolume centeredatthegridnodeand N isthetotalnumberofthemnodes.EachlocalnodeP maybesurroundedbysomeorallnodesofneighbornodes N , S , E , W , NE , SE , NW , SW dependingonthegeometryandlocationofnodeP. Considerageneralformoftheequationsinwhichshrinkagestrainandvolumetricex- ternalforcearepresentandcanbeincorporatedintooneterm,namely f inthegoverning equation, ¶s ij ¶ x j + f i = 0(3.1) Hereindexnotationhasbeenusedinsteadof x and y forthespatialdirections.The boundaryconditionscanbemoregeneralcomparedwith( 2.31 ),forexample,theymay includeaspvalueofthenonzeronormalstress.Inthiscontext,thestandardplane 31 stressrelationsbetweenstrainandstresswereused,giving s ij = E 1 + n # ij + E n 1 n 2 ( # kk ) d ij ,(3.2) whilethesmalldeformationrelation( 2.22 )stillremains.Initsindexnotationform,this termis # ij = 1 2 ¶ u i ¶ x j + ¶ u j ¶ x i ! .(3.3) IntegratingEq.( 3.1 )overeach w k gives ZZ w k ¶s ij ¶ x j + f i ! dxdy = 0.(3.4) Eq.( 3.4 )canalsobeinterpretedastheweakforminFEM,inwhichtheweightfunctions arechosenasHeavisidestepfunction.ApplyingGreen'stheorem,Eq.( 3.4 )becomes Z ¶w k s ij n j ds = ZZ w k f i dxdy ,(3.5) where n istheoutwardnormalunitvectorontheboundary.Thestencilsderivedforeach nodedependonfactorssuchas:theorderoftruncationerrorintheapproximationofthe spatialderivativesandthegeometryof w k andtheboundaryconditions.Forillustration purposes,considertheexampleofequibiaxialpullingandlet w k bethedomainindicated inFig. 3.1 ,surroundingnodePandhavingthreeneighboringnodesN,EandNE.Thus nodePisthelowerleftcornerofthecomputationaldomain.Thefouredgesofthesub- volume w k are g 1 , g 2 , g 3 , g 4 .Inthiscase,theshearstressiszeroandthenormalstresses havethevalue s 0 onthedomainboundary ¶ B ,andthusonsegments g 1 , g 4 . Forauniformmeshofdimension h ,thefouredgesallhaveequallength,whichis h 2 . On g 1 : s xx = s 0 , s xy = 0;on g 4 : s yy = s 0 , s xy = 0.ThecomponentofEq.( 3.5 )reads: Z ¶w k s 1 j n j dxdy = ZZ w k f 1 dxdy ,(3.6) 32 Figure3.1:ThesubvolumeatnodePanditsneighbornodes. inwhichtheoutwardunitnormalvectorsofthefouredgesare: ( 1,0 ) , ( 0,1 ) , ( 1,0 ) , ( 0, 1 ) ,therefore.Eq.( 3.6 ),using x and y notation,becomes Z g 1 s xx ds Z g 3 s xx ds + Z g 2 s xy ds Z g 4 s xy ds = ZZ w k f x dxdy .(3.7) On g 3 , s xx = s 0 , s xy = 0andon g 4 ,wehave s xy = 0, s yy = s 0 .Substitutingthese expressionsintoEq.( 3.7 )gives Z g 1 s xx ds + Z g 2 s xy ds Z g 3 s 0 ds 0 = 1 4 h 2 f x ,(3.8) where f x isthevolumeaveragedvalueof f x .Similarly,the y componentofthegoverning equationalongwiththeboundaryconditionsgives Z g 1 s xy ds + Z g 2 s yy ds 0 s 0 Z g 4 ds = 1 4 h 2 f y (3.9) FromEqs.( 3.2 )and( 3.3 ),weobtain 33 s xx = E 1 n 2 ¶ u ¶ x + n ¶ v ¶ y ,(3.10) s yy = E 1 n 2 ¶ v ¶ y + n ¶ u ¶ x ,(3.11) s xy = E 2 ( 1 + n ) ¶ u ¶ y + ¶ v ¶ x .(3.12) Nowwerequireanapproximationforthedisplacementderivatives.Thefollowing orderapproximationsareused: Z g 1 ¶f ¶ x ds = ¶f ¶ x j g 1 Z g 1 ds = 3 4 f f h + 1 4 f NE f h h 2 + O ( h ) ,(3.13) and Z g 2 ¶f ¶ y ds = ¶f ¶ y j g 2 Z g 2 ds = 3 4 f f h + 1 4 f NE f h h 2 + O ( h ) .(3.14) Notethaton g 1 , ds = dy andtheintegralof ¶f ¶ y isthetrueintegrandwhichcanbetaken as Z g 1 ¶f ¶ y ds = 1 4 ( f + f + f NE + f ) 1 2 ( f + f ) h 2 + O ( h ) = 1 4 ( f NE + f f f )+ O ( h ) . (3.15) 34 Similarly,on g 2 , ds = dx and Z g 2 ¶f ¶ x ds = 1 4 ( f + f + f NE + f ) 1 2 ( f + f ) + O ( h ) = 1 4 ( f + f NE f f )+ O ( h ) . (3.16) UsingtheaboveapproximationsforthederivativesofdisplacementsinEqs.( 3.8 )and ( 3.9 ),twostencilsfornodePare 3 8 ( u E u P )+ 1 8 ( u NE u N )+ n 4 ( v N + v NE v P v E ) + 3 8 1 n 2 ( u N u P )+ 1 n 2 1 8 ( u NE u E )+ 1 4 1 n 2 ( v NE + v E v N v P ) (3.17) = 1 4 f x h 2 + g h 2 , and 3 8 ( v N v P )+ 1 8 ( v NE v E )+ n 4 ( u NE + u E u N u P ) + 3 8 1 n 2 ( v E v P )+ 1 8 1 n 2 ( v NE v N )+ 1 4 1 n 2 ( u N + u NE u P u E ) (3.18) = 1 4 f y h 2 + g h 2 , where g = 1 n 2 E s 0 .Inthecaseofunequalbiaxialstresses,thevaluesof s 0 inthe g expressionintheandsecondstencilsshouldbereplacedbythevaluesofthepulling stressesinthe x and y directions, s 0 x and s 0 y respectively. Afterrearrangement,thestencilsfortheconsideredsubvolumeare: 35 8 > > > > < > > > > : ( 0 ) 1 8 + 3 8 1 n 2 1 8 + 1 8 1 n 2 ( 0 ) 3 8 3 8 1 n 2 3 8 1 8 1 n 2 ( 0 )( 0 )( 0 ) 9 > > > > = > > > > ; + 8 > > > > < > > > > : ( 0 ) n 4 1 4 1 n 2 n 4 + 1 4 1 n 2 ( 0 ) n 4 1 4 1 n 2 n 4 + 1 4 1 n 2 ( 0 )( 0 )( 0 ) 9 > > > > = > > > > ; = 1 4 f x h 2 + 1 2 gh ,(3.19) 8 > > > > < > > > > : ( 0 ) 1 4 n + 1 4 1 n 2 1 4 n + 1 4 1 n 2 ( 0 ) 1 4 n 1 4 1 n 2 1 4 n 1 4 1 n 2 ( 0 )( 0 )( 0 ) 9 > > > > = > > > > ; + 8 > > > > < > > > > : ( 0 ) 3 8 1 8 1 n 2 1 8 + 1 8 1 n 2 ( 0 ) 3 8 3 8 1 n 2 1 8 + 3 8 1 n 2 ( 0 )( 0 ) SE ( 0 ) 9 > > > > = > > > > ; = 1 4 f y h 2 + 1 2 gh .(3.20) Herethefollowingconventionisusedtosimplifythenotation.ThematricesinEqs. ( 3.19 )and( 3.20 )containtheparametersassociatedwith u ,thesecondwith v andthe positionsoftheparametersinamatrixindicatethenode.Inparticular: 8 > > > > < > > > > : NWNNE WPE SWSSE 9 > > > > = > > > > ; .(3.21) Inasimilarway,thestencilsforthesubvolumeatthelowerrightcornerofthesample whichinvolvefourlocalnodesP,N,W,NW,are 36 8 > > > > < > > > > : 1 8 + 1 8 1 n 2 1 8 + 3 8 1 n 2 ( 0 ) 3 8 1 8 1 n 2 3 8 3 8 1 n 2 ( 0 ) ( 0 )( 0 )( 0 ) 9 > > > > = > > > > ; + 8 > > > > < > > > > : n 4 1 4 1 n 2 n 4 + 1 4 1 n 2 ( 0 ) n 4 1 4 1 n 2 n 4 + 1 4 1 n 2 ( 0 ) ( 0 )( 0 )( 0 ) 9 > > > > = > > > > ; = 1 4 f x h 2 1 2 gh ,(3.22) and 8 > > > > < > > > > : 1 4 n 1 4 1 n 2 1 4 n 1 4 1 n 2 ( 0 ) 1 4 n + 1 4 1 n 2 1 4 n + 1 4 1 n 2 ( 0 ) ( 0 )( 0 )( 0 ) 9 > > > > = > > > > ; + 8 > > > > < > > > > : 1 8 1 n 2 + 1 8 1 8 1 n 2 + 3 8 ( 0 ) 3 8 1 n 2 1 8 3 8 1 n 2 3 8 ( 0 ) ( 0 )( 0 )( 0 ) 9 > > > > = > > > > ; = 1 4 f y h 2 + 1 2 gh .(3.23) Fortheupperrightcornerofthesample,thestencilsare 8 > > > > < > > > > : ( 0 )( 0 )( 0 ) 3 8 1 8 1 n 2 3 8 3 8 1 n 2 ( 0 ) 1 8 + 1 8 1 n 2 1 8 + 3 8 1 n 2 ( 0 ) 9 > > > > = > > > > ; + 8 > > > > < > > > > : ( 0 )( 0 )( 0 ) n 4 + 1 4 1 n 2 n 4 1 4 1 n 2 ( 0 ) n 4 + 1 4 1 n 2 n 4 1 4 1 n 2 ( 0 ) 9 > > > > = > > > > ; = 1 4 f x h 2 1 2 gh ,(3.24) 37 and 8 > > > > < > > > > : ( 0 )( 0 )( 0 ) 1 4 1 n 2 + 1 4 n 1 4 1 n 2 1 4 n ( 0 ) 1 4 1 n 2 + 1 4 n 1 4 1 n 2 1 4 n ( 0 ) 9 > > > > = > > > > ; + 8 > > > > < > > > > : ( 0 )( 0 )( 0 ) v 3 8 1 n 2 1 8 1 8 1 n 2 + 3 8 ( 0 ) 1 8 1 n 2 + 1 8 1 8 1 n 2 + 3 8 ( 0 ) 9 > > > > = > > > > ; = 1 4 f y h 2 1 2 gh .(3.25) Fortheupperleftcornerofthesample,thestencilsare: 8 > > > > < > > > > : ( 0 )( 0 )( 0 ) ( 0 ) 3 8 3 8 1 n 2 3 8 1 8 1 n 2 ( 0 ) 1 8 + 3 8 1 n 2 1 8 + 1 8 1 n 2 9 > > > > = > > > > ; + 8 > > > > < > > > > : ( 0 )( 0 )( 0 ) ( 0 ) n 4 1 4 1 n 2 n 4 1 4 1 n 2 ( 0 ) n 4 + 1 4 1 n 2 n 4 1 4 1 n 2 9 > > > > = > > > > ; = 1 4 f x h 2 + 1 2 gh ,(3.26) and 8 > > > > < > > > > : ( 0 )( 0 )( 0 ) ( 0 ) 1 4 1 n 2 + 1 4 n 1 4 1 n 2 1 4 n ( 0 ) 1 4 1 n 2 + 1 4 n 1 4 1 n 2 1 4 n 9 > > > > = > > > > ; + 8 > > > > < > > > > : ( 0 )( 0 )( 0 ) ( 0 ) 3 8 1 n 2 3 8 3 8 1 n 2 1 8 ( 0 ) 1 8 1 n 2 + 3 8 1 8 1 n 2 + 1 8 9 > > > > = > > > > ; = 1 4 f y h 2 1 2 gh .(3.27) 38 Tofullyderivethestencilsforallofthenodesinthegrid,manydifferentcasesmustbe considered,eachcasecorrespondingtothewaythematerialintheneighborhoodofthe node P isremovedduetodamage.SuchdetailedresultsarepresentedintheAppendix forinterestedreaders.Inthenextsection,theFEMwillbediscussed.TheFEMreduces theworkloadofdevelopingstencilsforeachnode. 3.2FiniteElementimplementation TheFiniteElementmethod,duetoitsrigorousmathematicalfoundation,hasbeen usedextensivelytoproducesomenumericalsolutionsforsomeproblemsinsolidme- chanicsandrecently,inmechanics.Theisotropictriangularmeshisgeneratedbythe MATLABfunction mesh 2 d . m whichusetheDelaunayalgorithmtominimizetheband- width.Boththeinitialboundaryvalueheattransferandthestressproblemsdescribedin Sections 2.2 and 2.3 arenumericallyresolvedusingFEM.TheformulationoftheFEMfor thelatterisoutlined.TosimplifythenotationintheFEMformulationofthestressbalance equation,theLameparametersareusedinsteadofYoung'smodulusandPoisson'sratio andforplanestresscondition,theirrelationsaregivenby l 0 = E n 1 n 2 , m = E 2 ( 1 + n ) . Thustheconstitutiverelationsbetweenstressandstrain,Eqs.( 2.19 ),( 2.20 ),( 2.21 ),canbe writtenusingindexnotationas s ij = l 0 ( # 11 + # 22 ) d ij + 2 m# ij + s 0 d ij ,(3.28) where s 0 = g ( 1 n ) r r 0 r 0 ,(3.29) 39 andtherelationsrelatedstraintodisplacementaregivenbyEqs.( 2.22 )or( 3.3 ).Inthis work,theorderlinearFEMonatriangularmeshisutilized.Thelineartriangularin referencecoordinates ( x , h ) onwhichtheintegraloftheweakformisperformedare: 0 x 1, 0 h 1 x ,(3.30) andtheshapefunctionsinthiscoordinatesystemaregivenby: j 1 ( x , h )= x h + 1, j 2 ( x , h )= x ,(3.31) j 3 ( x , h )= h . Theseshapefunctionsprovidealineartransformationthattransformthetriangleinthe domain( 3.30 )intoatriangle w e withvertices ( x k , y k ) , k = 1,2,3inthematerialdomain, whichisgivenby x = å k x k j k ( x , h ) , y = å k y k j k ( x , h ) ,(3.32) andanapproximationofthedisplacements u and v bytheirnodalvalues,whichare 40 unknown u = å k u k j k ( x , h ) , v = å k v k j k ( x , h ) .(3.33) TheweakformofEq.( 3.28 )canbeobtainedbymultiplyingthePDEswithweightfunc- tionsandintegratingthemoverthecomputationaldomain, B .UsingGalerkin'sap- proach,thesesixindependent2Dweightfunctionsarechosentobetheshapefunctions inEq.( 3.31 ) 0 B @ j 1 0 1 C A , 0 B @ 0 j 1 1 C A , 0 B @ j 2 0 1 C A , 0 B @ 0 j 2 1 C A , 0 B @ j 3 0 1 C A , 0 B @ 0 j 3 1 C A (3.34) Theintegralovereachelement w e istransformedintotheintegraloverthelocal triangle( 3.30 )using( 3.32 )withtheJacobianofthetransformation( 3.32 )givenby J ( x , h )= ¶ x ¶x ¶ x ¶h ¶ y ¶x ¶ y ¶h . TheFEMmeshusedthroughoutthesenumericalsimulationscontainsaround N = 1 e 5nodes.Thethermalproblemhasonedegreeoffreedom, dof = 1,whichisthetem- peraturepereachnodewhilethestressproblemhastwo, dof = 2,whichare u and v . Thusthetotalnumberofunknownsineachproblemareoftheorder1 e 5and2 e 5,which isarelativelylargenumbercomparedtothecapacityofatypicalpersonalcomputer.Thus itisdesirabletoparallelizethecodeandScaLAPACKlibraryisusedforthatpurpose. Theglobalstiffnessmatrix G isassembledandstoredincyclicorderbymultipleproces- sorsoperatinginparallel.Whenanelementisremovedfromthecomputationaldomain becauseitsmaximumprinciplestressreachesthethresholdvalue, G isupdatedbythe 41 processorsthatareinvolvedintheelement.Duetothepiecewiseshapefunctions,each nodeisonlyconnectedtoitsnearestneighbor,thus G isabandedmatrixwhoseband- widthdependsonthenodenumberingtechnique.Thesystemoflinearalgebraicequa- tionsissolvedateachtimestepbytheparallelLUfactorizationfunctionofScaLAPACK [ 119 ]whilesomesteps,suchascalculatingtheprinciplestressesforallelements,remain serial.ScaLAPACKisalsocalledtowriteoutputtoinsynchronousorder. 3.3Temperatureanddensity InSections 3.3 and 3.4 ,thenumericalsimulationsontherectangulardomain L H = 5 cm 2 cm willbepresentedforonesetofparametervalues.Thematerialproperties usedhereinareinthecharacteristicrangeforagenericcharring,rubber-likematerial. RepresentativevaluesforthepropertiesofsuchmaterialscanbefoundinRefs.[ 120 ], [ 121 ]and[ 122 ].Thefollowingnumericalvaluesareusedforthethermal,pyrolysisand elasticparameters: l = 4.0 10 7 m 2 s 1 q 0 / k y = 6.0 10 5 K / m r c / r 0 = 0.3 T a = 9375 K A = e 31.25 s 1 (3.35) n = 0.45 g = 1/3 T 0 = 300 K s c / E = 1/30. AllnumericalresultspresentedinthistheisareproducedbytheFEMwithlinear 42 triangularmeshandmeshsize h e ofapproximately0.01 cm .Thetemperatureis bytheunsteadyheatconductionEq.( 2.10 )alongwiththeinitialandboundary conditionsgivenbyEqs.( 2.25 ),( 2.26 ),( 2.27 ),( 2.28 ),( 2.29 ),( 2.30 ).Forbothproblems,the timestepischosentobe1second,howevertheresultsareonlywrittentooutputat eachcertaintimeintervalbecauseofthelimitedstorageability.Heretheyarewrittenout inevery50 s . Inthismodel,theheatcondition( 2.30 )isappliedfrom t = 0throughthewhole simulation,whichmeanstheheatiscontinuouslysuppliedandtimeindependent. Moreover,thereisnocoolingmechanismsuchasconvectionorradiationcarryingheat awayfromthesurfaces,thusthetemperaturekeepsrisingwithtimeuntilreachingequi- libriumorsteadystate.Thetemperatureattainsitshighestvalueatthecenteroftheheat location.Thiscanbeseenfromasequenceoftemperatureplotsat50,1000,6000 s in Fig. 3.2 .Thesamplelowersurfaceisinsulated, ¶ T ¶ y = 0,Eq.( 2.27 ),sothetemperature contourisverticallytangenttothissurface.Thetwolateralsidesaremaintainedatthe initialtemperature T 0 ,Eq.( 2.25 ).Thisisthelowesttemperaturevalueinthedomain. Thedistribution q alongtheverticalmiddlelineofthesampleat t = 100,1000,2000, 3000,4000,5000,6000 s canbeseenfromFig. 3.3 . Initially,themasslossrateiszeroornegligiblysmallduetolowtemperature,thusthe densityisuniformlyatitsoriginalvalue.Afterthat,thesamplepyrolyzes,beginningat theuppersurface.Adensityof99.95%oftheoriginalvalueisconsideredastheonset ofpyrolysis.Thishappensat t = 18 s .Thesampledensityalwayshasitslowestvalue atthelocationofthehighesttemperature,thecenteroftheheatedregion.Thechar valueisattainedthere,atapproximately116 s ,seeFig. 3.4 (a).Beforethelowerinsulated surfacestartspyrolyzingat4500 s ,thecharlayerregiongrowsradiallybecausetheheat islocalized.Thecharregionavoidsthetwocoldwallsbecausethelowtemperature 43 Figure3.2:Thedimensionlesstemperature q =( T T 0 ) / T 0 at t =50,1000,6000 s from toptobottomes.Theleftandrightsidesarecoldwallswhilelowerwallisinsulated. Thesampleisheatedattheuppersurfacebyaheattothecenterregionover thelength l = 0.1 L inthoseregionsproducesanegligiblemasslossrateintheArrheniusEq.( 2.2 ). 44 Figure3.3:Thedistributionofthedimensionlesstemperature q alongtheverticalmiddle lineofthesamplewhere x = 0.5atdifferenttimesasindicatedinthelegend,inwhich x = x / L and y = y / L Thelocationfurthestfromthecenteroftheheatthathaschardensity r = 0.3iscalled thecharfrontandthelocationwherepyrolysishasnotyetstartedthatisclosesttothat centeristermedthepyrolysisfront.Thesefrontsarecurvesinthis2Dproblemandcan beseenfromFig. 3.5 .Furthermore,thedensitygradientisappreciableonlyinanarrow regionthatseparatesthecharredfromtheuncharredregions,orbetweenthetwofronts. Overthissmalldistance, d ,thedimensionlessdensity r ,as r = r / r 0 ,dropsfrom theuncharredvalue r = 1.0tocharredvalue r = 0.3(darkbluecolortowhitecolorin Fig. 3.5 ).Incalculation,thevalue r = 0.99insteadof r = 1.0istakenfortrackingthe pyrolysisfrontbecauseofpracticalpurpose.Themagnitudeofthedensitygradientmay beapproximatedby ( r 0 r c ) / d ,where d assumesdifferentvaluesatdifferentlocations duringtheheatingandpyrolyzingprocess. Theevolutionof r and d atthemid-verticallineofthesampleisplottedinFig. 3.6 , wheretime t isrescaledwiththecharacteristicheatconductiontime, t hc = H 2 / a y .A moresystematicforchoosingthisvaluewillbegiveninSection 6 .Forthe parametersusedinthissimulation, t hc = 1000 s .At t / t hc = 2,thecharfrontisstillfar 45 Figure3.4:Plotsof r fromlefttoright,toptobottomoftheecorrespondsto t =100, 300,1000,1500,2000,3000,4500,6000 s ,respectively.Thecharregiongrowsradiallyfrom thecenteroftheheatedregion.Closeto t =6000 s ,charregionreachestheinsulatedlower surfacebutavoidsthecoldsidewalls. Figure3.5:Charfrontandpyrolysisfrontareindicatedbycurvedarcs.Theirseparation d atonelocationisindicatedbythe ! . fromlowersurface.Uptothistime,thecurve d vs. t / t hc iswellbythesquareroot function t 1/2 sothedensitygradientdecreasesas t 1/2 .Ananalyticalderivationofthis resultforaonedimensionalpyrolysismodelispresentedin[ 11 ]. 46 Figure3.6:(a)Thedimensionlessdensity r = r / r 0 atthemid-verticalline( x = 0.5, where x = x / L )atdifferenttimes.Itdecreasesfromuncharred( r = 1.0)tothechar value( r = 0.3)overthedistance d betweenthecharredanduncharredregions.(b)The evolutionofthelocationsthat r = 0.3,0.5,1.0and d .The d vs.timecurvewell toasquarerootfunction. Underthesimplifyingassumptionthatthesampleshallnevercrack,onemaycalcu- latetheupperlimit s m thatthemaximumprinciplestresswillattainoverthecourseofany heatingtime. s m varieswithparametervaluesexceptthecrackingthreshold s c . Forthesetofparametersusedinthissimulation,thislimitvalueis s m = 0.18666 E ;when s c > s m ,nocrackswillforminthesamplefromheating.When s c < s m andthesample fracturesinresponsetotheaccessiblemaximumprinciplestresscriterion,thesamplema- terialisdepletedbythesequentialremovalofelementsfromthecomputationaldomain. Asadirectconsequenceoftheremovalofelements,thestressiscorrespondingly Itisgenerallyconcentrated(enhanced)nearthecracktip(damagedelement). Priortocracking,thestressinthiscaseisqualitativelysimpletounderstand.Fig. 3.7 showsthemaximumprinciplestress s 1 at t =150,1000,3000 s whenthesampledoes notcrack.Itcanbeseenthatthelocationofhighmaximumprinciplestress, s 1 ,correlates withthelocationofthehighdensitygradient.Furthermore,themaximumvalueof s 1 decreasesfromthetime150 s totheendofthesimulation. 47 Figure3.7:Maximumprinciplestress s 1 attimes t =150,1000,3000,5500 s whenthesample tensilestrength s c ishigherthan s m .Insuchacasethesampledoesnotdevelopcracks. 3.4Generalbehaviorofcrackevolutionandmorphology Theprevioussectiondiscussedtheevolutionofthetemperature,densityandstress withoutcracksforonesetofparametersgivenbyEq.( 3.35 )except s c ,inwhich thecondition s c > s m istherelevantassumption.Inthissection,thesamedensity isusedforshrinkagestraininthestressproblembutcracksarenowallowedtodevelop byusingvalueof s c thatissmallerthan s m asinEq.( 3.35 ).Thegeneralbehaviorfor theevolutionandmorphologyofthecrackswillbediscussedhere.Fig. 3.8 showsthe distributionofthemaximumprinciplestress s 1 andtheevolutionofcracksupto t =6000 s atseveraltimesasindicatedinthee. Basedontheevolutionoftemperature,densityandcrackmorphology,theprocesscan bedividedintostages.Thesestagesare:(a)inertheating(b)pyrolysis,notcharred and,crackinitiation,(c)slightlycharred,initiationdominant,(d)halfcharredand fastpropagation,(e)almostfullycharred,deceleratedpropagation.Inthestage(a), thesampletemperaturerisesbutistoosmalltoproduceanappreciablemassof volatiles.Thisisthesameasthestageof[ 11 ].Theevidenceforthisstageisbasedon valuesofthesurfacedensityforwhich r = 0.9995whichisattainedat t = 18 s .The 48 Figure3.8:Maximumprinciplestress s 1 .Plotsfromlefttoright,toptobottomcorrespond to t =75,300,1000,1500,2000,3000,4500,6000 s ,respectively.Attheendofthesecond stage, t =75 s ,thecrackinitiates. stagethenistakentoevolvefrom t = 0to t =18 s .Thetemperaturedistributionat t =18 s is seeninFig. 3.9 .Followingthisstageisstage(b),pyrolysisbeginsbutthesampledensity isstillwellabovethecharvalue.Thecrackingthresholdhasnotyetbeenattainedbythe maximumprinciplestressduetolowdensitygradient.Thedensitygradienteventually attainsasufhighmagnitudeforthecracktonucleateat t =75 s ,whichmarks theendofstage(b).Atthistime,thelowestdensityvalueis r = 0.8345. Thethirdstage(c)ischaracterizedbyadensitywhosegradientdecaysas t 1/2 andcon- tinuestowardthenextstage,asmentionedintheprevioussection.Cracksinitiatefrom theheatedsurfaceandpropagateradiallyoutward.Thedensityattainsthechar value r = 0.3when t = 116 s ,seeFig. 3.10 .Eventually,initiationactivityisdimin- 49 Figure3.9: q attheendofthestage t = 18 s when r = 0.995atthemiddleofthe heatedregionontheuppersurface. ishedandreplacedbycrackelongationandbranching,whichindicatestransitiontothe nextstage(d).Attheendofstage(d)thedensityatthelowersurfacealsoattainsthe charvalue.Crackinitiationisnolongerobservedandtheexistingcrackspropagateata slowerpace.Thesecracksnowintersecteachother,formingloopsandnetwork-likepat- terns.Thetimesforeachstagearethefollowing:(a)from0 s to18 s ,(b)from19 s to75 s ,(c)from76 s to116 s ,(d)from117 s toabout4500 s ,(e)fromabout4500 s onward. Figure3.10: r attheendofthethirdstage t = 116 s whentheuppersurfacestartscharring. Awayfromthetwocoldwalls,thedensitygradientattainsmaximummagnitudedur- ingthestage(b)andthengraduallydecreaseinstrengthovertimeasthecharfrontmoves fromsurfaceintothesampleinteriorand d grows.Thisisduelargelytothediffusive natureofheatconduction.Themaximumprinciplestressattainedinthematerialalso decreases,whichexplainswhyinitiationhappensatthesurfaceandthenmovesinto thesamplealongwiththeregionofhighdensitygradient.Fig. 3.11 showsanexample 50 ofthisbehavior.Laterintheprocess,densitygradientmagnitudedecreasesandthisre- ducesthemaximumattainableprincipalstress,wherebyfewnewcracksareabletoform. Figure3.11:Maximumprinciplestress s 1 at300 s and1000 s .Initiationsitesindicatedin theplotsareatthesampleinteriorwherethelocaldensitygradientconcentrates.Initia- tionactivitystopsatapproximately400 s . Theoveralltrendofpropagationisdirectional,followingthecharfrontfromthesur- facetotheunburnedregion.Cracksdevelopradiallyandperpendicularto,andmovein advanceof,thecharfront.Itisknownthatcracksadvancesinthedirectionobeyingthe principleofenergyminimizationandinthedirectionofmaximumtangentialstress[ 123 ]. Thecorrelationbetweencrackpropagationandthedensityisseenmoreeasilyby plottingthecrackdistributionoverthedensityseeFig. 3.12 .Hereitisrecalledthat inthismodel,theevolutionofdensityandtemperaturearenotaffectedbythepresence ofcracks. Itcanbeseenfromtheplotsforearlytimesthatthecrackpatternexhibitsahierarchi- calbehaviorsuchthatlongandshortbranchesalternatewitheachotherduringthisstage. 51 Figure3.12:Plotsof r insamplealongwithcrackingpatternat t =75,300,1000,1500,2000, 3000,4500,6000 s .Crackinginititatesat75 s .At t = 100 s ,crackshavealreadydeveloped whiletheuppersurfacehasnotbeencharredyet.Thepyrolysisfront,followedbythe charfront,isalwaysbehindthelongestcracks.Atalatertime,around4500 s ,thelower surfacestartstopyrolyze;crackspacinginthemiddleregionnearthelowersurfacegets larger.Thisisalsothelocationwhere d islargeandthedensitygradientissmall.Cracks advanceindirectionsthatareperpendiculartothepyrolysisfront. Thiselegantpatternissimilartothedistinctiveperiodicdoublingpatternrecognizedin quenchingandcoldshockexperimentsby[ 67 ]inrectangularorin[ 126 ]circularsample. Thishierarchyismoreapparentinstage(c)ortheearlyofstage(d)partlybecausethe crackshavenotyetintersectedwitheachother(astheydoinlatestage(d)),asseenin Fig. 3.13 .Unlikecrackingarraysinquenchedplateswhichinitiateatthesametimeand formtheperiodicdoublingpatternsbyclosingeveryothercrack,thecracksinthissim- ulationcaninitiateatdifferenttimesandmostcontinuetopropagatethroughtheendof thesimulationunlesstheyintersectwithoneanother. 52 Figure3.13:Thehierachicalstructureofthecrackpattern.Themaximumprinciplestress s 1 at3000 s .Thelong(indicatedbyl ! )andtheshort(s ! )branchesalternateeach other. Formostofthetime,themaximumprincipalstress s 1 atcracktipislargerthanthe criticalcrackingvalue s c ,thuscracktipswillactivelypropagate.Crackinitiation,which happensduringstage(c)andinearlystage(d)islessfavoredthancrackpropagation.As aconsequence,cracksusuallycomprisemanywelldevelopedbranchesinsteadofshort andisolatedes. Earliercracksmodifythestressaroundnewlyemergedcracksandviceversa.From thesequencesofcrackimages,itisobservedthatwhenonecrackpropagates,thecrack initsneighborhoodceasesmovingtemporarilyduringwhichtimethetensilestresses arounditstipincreasebeforeitonceagaincontinuesinitsforwardmotion.Thismaybe describedasamutualunloadingbehavior,withthecracksrelievingtotalenergyinthe vicinityoftheirneighboringcracksinanalternating,periodicmanner.Thisexplainsalso whythesecrackspropagateinadiscreteratherthaninacontinuousmanner. Aftertheinitiationandpropagationperiod,attheendofstage(d),theshorterbranches eventuallyjointhelongeroneswhichalreadyhavecurvedahead.Thejoiningoftwo branchesformsaloopwhichisanisolatedfragment.Furthermore,theseloopshave shapesofpolygonsthatareelongatedtowardtheuncharredregion.Analysisoftheloop patternisprovidedinchapter 4 .Crackstendtointersecttheexistingcracksorafreesur- faceatarightangle.Thistendencyisexplainedbyusingtheprincipleofmaximumstress 53 releaseandcrackpropagation,asproposedbyLachenbruch(1962)[ 127 ].Incontrastto crackintersectionthatjoinstwocracksistheprocessofcrackbranching,whichoccurs eitherbykinking(asharpturnthatcanraisethestress)orbifurcation(splittingofthe cracktips).Forexamplesofjunctions,seeFig. 3.14 ,inwhichthreeintersectionsatright anglesaremarkedbytheletterTinred.Moreover,inFig. 3.15 forthecrackpatternat 700 s ,onecracktipisjustabouttosplit.At1200 s ,itbecomestwoactivebranches.The triplejunctionanglesformedbythreesegments:oneflmotherflcrackanditstwoflchil- drenflbranches,deviatesfrom ( 120 o ,120 o ,120 o ) .Thisfactrtheanisotropicnature ofthedrivingFig. 3.16 showexamplesofthenucleationatthekinkofachildcrack. Thesetypesofjunctionsarediscussedinastudyaboutjunctionformationindesiccation cracking[ 55 ]. Figure3.14:Maximumprinciplestress s 1 at6000 s .Twocrackintersectionsatrightangle aremarkedbytheletterTinred. Itisworthwidetonotethatwhilethetemperature(Fig. 3.2 ),density(Fig. 3.4 ) andstresswithoutcracks(Fig. 3.7 )aresymmetricaboutthemidverticalline x = 0.5, thecrackpatternsandthusstresswithcracks(Fig. 3.8 )arenot.Thebreakingof symmetryiscausedbyunsymmetryofthetriangularmeshwhenelementsareremoved. However,thecrackpatternsontheleftandrightpartsofthesamplearestatisticallysim- ilar.Thisissueisdiscussedalittlefurtherintheappendix,section C .Moreover,itcanbe seenfromtheappendixsection D ,thatfurtherreofmeshdoesnotleadtostatis- 54 Figure3.15:Maximumprinciplestress s 1 at700 s and1200 s .Thecracktipindicated startssplittingat t = 700 s .At1200 s ,ithassplitintotwobranches. ticallydifferentcrackpatterns.Thusthecurrentmeshusedissuftoresolve detailsofcrackevolution. Insummary,thecrackpatternevolvesthroughdifferentcharacteristicstagesofcrack formationandpropagation,thesebeingstages(c),(d)and(e)oftheoverallprocess.In addition,eachstageischaracterizedbyadominantphysicalmechanism.Inparticular,the crackpatternsaregeneratedthroughtheactionoftwocompetingmechanismsofcrack evolution,namelybranchingandjoining.Inthebroadercontextofphysicallyinduced patternformation,theformer(branching)characterizesthedevelopmentofahierarchi- calnetwork,whichistypicallycausedbythetransportofmacroscopicquantities,suchas water,orcellinawater-channelsystem,leafveins,bloodveins,ortreebranches, electricalthermalenergy,etc.Thelatter(joining)generatesanetworkduetotheco- alescenceofbranchesandistypicallygovernedbyaprincipleofenergyminimization,as foundinmudcracks,glazes,etc.Whereasthemechanismisdirectionalanddrives thecrackpatternintheearlystagesofthesesimulations,theotherisisotropicanddom- inantinthelaterstages.Inthiswork,thefiveinsflandpathwaysforthemechanism 55 Figure3.16:Maximumprinciplestress s 1 at3800,4000,4500 s .Junctionsformedeitherby thenucleationofachildcrackorbykinkingareindicatedwithredarrows. (branching)arecreatedbythestressinconjunctionwiththeapplicationofthemax- imumprinciplestresscriterioninthematerialandarenot,asintheprocessesmentioned above,pre-arrangedeitherbiologicallyormaterially. 56 CHAPTER4 MORPHOLOGICALANALYSISANDQUANTIFICATIONOFCRACK MORPHOLOGY Avarietyofmethodshasbeenusedfordetectingcracks,suchasmethodsofopticalorul- trasonic,imaging.Themacrocrackimagecanbecapturedbyacamerawhereaselectrical resistancetomography(ERT)andscanningelectronmicroscopy(SEM)areusedformicro sizecracks.ImagingisusuallyfollowedbyimageanalysisforfurtherIn thischapter,somefundamentalconceptsofmathematicalmorphologyandimageanaly- sisrelatedtoanalyzingcrackingpatternsarepresented,followedbytheirapplicationsto thepatternsthataregeneratedinthisstudy. IntheFEMtriangularmesh,whentheaveragedprincipalstressofanelementexceeds athresholdvalue,theelementisremovedfromthecomputationaldomain.Thischange ispermanent.Thehistoryoftheremovalprocessisstoredinacrackpatternvariable C byassigningthevalueofthevariableattheremovedelementlocationequaltothetime stepatwhichitwasremoved.Fromthisvariable,thecrackpatternateachtimestep isconstructed.Inparticular,extractingthecrackpatternattime t = t c from C requires searchingalllocationsatwhich C attainsvalueslessthan t c .Thesetofthesepointsforms thecrackpatternat t c .Thiswayofstoringalsofacilitateskeepingtrackofcrackorders, whichcanbeusedforassigningsegmentranks(asdiscussedlater). Analysisofthecrackpatternsisperformedonarectangulargridofpixels N x N y ,in which N x isthenumberofpixelsinthehorizontaldirectionand N y isintheverticaldi- rection.SincetheFEMmeshistriangular,itneedstobemappedontotherectangular mesh.Lettheinteger N denotetheratioofdimensionsbetweentherectangularmesh usedforimageanalysisandthetriangularFEMmesh.Thecentroidofeachtriangularel- ementiscalculated.When N equalsunity,thepairsofcentroidcoordinatesarerounded offtothenearestrectangularmeshgridpoints.Ingeneral, N > 1andthecentroidcoordi- 57 natesaremultipliedby N andthenroundedofftothe N nearestrectangulargridpoints. Therearedifferenttypesofpixelgrids,usuallyfourpointoreightpointconnectivityma- trix.Inthiswork,thelatterisused.Eightpointconnectivityconsidersthatanylocalnode isrepresentedbyapairofintegers ( i , j ) ,1 < i < N x ,1 < j < N y thatissurroundedby eightneighbornodes: ( i + 1, j ) , ( i 1, j ) , ( i , j + 1 ) , ( i , j 1 ) , ( i + 1, j + 1 ) , ( i + 1, j 1 ) , ( i 1, j + 1 ) , ( i 1, j 1 ) . Morphologicalfunctionsworkonbinaryimageswhichhaveonly0(background)and 1(foregroundorobject)pixels.Ifthecrackingpatternisobtainedfromacoloredpicture, theimagewillbethresholdedtoagrayscaleandthentoablackandwhiteimage.Al- thoughnotuseddirectlyinthiswork,therearesomeprimarymorphologicalfunctions (level0)usedtoconstructfunctionsofhigherlevelsinimageanalysis.Examplesofsuch primaryfunctionsarethefunctions erode,dilate,open,close inwhich dilate addssurround- ingpixelstoobjectpixelsand erode isthedilationofthebackground.Thefunction open is erode followedby dilate while close is dilate followedby erode .Thesefunctionsarecentered byastructuringelementwhichisamatrixcontaining0and1withalocationindicating thepixelwhichitactsupon.Thesefunctions,alongwiththenexthigherlevelones,can befoundasbuilt-infunctionsinMATLAB. Themorphologicalfunctionusedinthisworkis bwmorph . m of MATLAB which belongstolevel1.Aftertransformingfromthetriangularmeshtotherectangularmesh, thecrackpatternisthinnedtoaonepixelthicknessby bwmorph . m withtheoptionflskele- tonized.flBesidesflskeletonizedfl, bwmorph . m makesotheroptionsavailable,suchas ingbranchingpoints,endingpoints,leavingtheoutlineofashape,removingisolated pixels,thickeninganimage,...etc.Nevertheless,forthisworkthereisstillaneedfor constructingadditionalfunctionswhichareconsideredatlevel2. 58 4.1Someimageanalysisconceptsandalgorithms Assumingthecrackpatternhasasinglepixelthickness,thereareimportantconcepts inquantifyingourcrackmorphologythatneedtobeForanypixelintheblack andwhiteskeletonizedimage,theeightpointconnectivitymatrixprovidesinformation aboutthetotalnumberofsurroundingpointsandtheirpositionsrelativetothepoint beingconsidered.Toeliminatetheorientationdependenceoftheconnectivitymatrixon thecoordinatesystem,theangularinsteadoftheeightpointconnectivitymatrixisused. SomeauxiliaryconceptsrequiredforthisThepixelrectangulargridissetof allpixels ( i , j ) inwhich1 < i < N x ,1 < j < N y withorigin O , i = 1, j = 1.Thehorizontal axisvector ~ Ox pointsfromlefttoright.Anytwopixelsspecifyavectorinwhichthe counterclockwisedirectionalangle a itmakeswith ~ Ox 0 o a 360 o .(Two directionalangleshaveequalvaluesiftheirdifferencedividesasamultipleof360 o ).The subscript Ox impliesthatvector ~ Ox formsonesideoftheangle.Foreachpixel ( i , j ) ,theset ofall n vectorsinwhichthevectorheadsareneighborsof ( i , j ) intheconnectivitymatrix andvectortailatthepixelconsideredis ( ~ v 1 , ~ v 2 ,..., ~ v n ) .Thissetdividestheplaneinto n regions.The n vectorsarearrangedinascendingorder,meaning a 1 Ox < a 2 Ox < ... < a n Ox . Thesetof n anglesformedby n vectors: a = a 1 , a 2 ,..., a n 1 , a n ,satisfyingthecondition a j = a j + 1 Ox a j Ox ifj 6 = n , a n = 360 o n 1 å i = 1 a i ,(4.1) iscalledtheangularconnectivitymatrix. Asanexample,considerthe(foreground)pixel ( i , j ) inwhichitseightpointconnectivity matrixhastwoother(foreground)pixels ( i , j + 1 ) and ( i + 1, j + 1 ) .Inthiscase, n = 2 andtwovectors ~ v 1 =( 0,1 ) , ~ v 2 =( 1,1 ) centeredat(i,j)areformed,theirtwodirectional angleswith Ox being a 1 Ox = 90 o and a 2 Ox = 45 o .Inascendingorder, a 0 1 Ox = 45 o and a 0 2 Ox = 90 o .Thus a at ( i , j ) is ( 45 o ,315 o ) accordingtoEq.( 4.1 ).Ingeneral,sinceanypixel canhaveuptoeightneighboringpixelsinaneightpointconnectivitymatrix,thereisthe 59 possibilitythat a haseightelements,eachbeingmultipleof45 o .However,inaskele- tonizedimage,thiswouldnothappen.Dependingontherelativearrangementbetween pixelsinskeletonizedimage,onepixelcanbeeither:anendpoint,ajunctionbetween threeormoresegmentsoramiddlepointofacracksegment. Knowingtherelativepositionofanypixelinaskeletonizedimagepattern,thecrack pathfromanycracktiporbranchpointtothenearestbranchpointoranothertipcan bestoredinavariablebymarkingthecoordinatesofallthepixelslyingonthepathina sequentialorder.Thisistermedacracksegment.Asegmentisanisolatedcrackifitcon- nectstwocracktips(endpoints).Wewillshowlaterthatthiscanalsobefoundbyusing bwboundariesMATLAB function.Alevel2functioniswrittentocracksegmentsin anynetworklikepattern,notlimitedtotheonesgeneratedinthisstudy.Thealgorithm forcracksegmentsispresentedbelow: a)Findtheset P ofallendpointsandjunctionsinaskeletonizedimage b)Foreachpointin P ,theuntraveledpathtoothernearestendpointorjunction,ina wayapenciltraversesthroughallpixelswithoutbeingliftedfromthepaper.Recordthis pathasonesegment.Markthepathasfltraversedfl. c)Repeatb)untilthereisnoremaininguntraveledpath. Toaidinvisualization,eachsegmentiscoloredrandomlybyassigning R , G , B random valuesthatvarybetween0and225tocreatarandom ( RGB ) tripletforeachsegment. Thesegmentlengthcanalsobeestimated.Therearedifferentmethodsofestimation. Thesimplestoneapproximatesthetotalnumberofpixelsineachsegmenttoitslength, regardlessoftherelativearrangementofthepixels.Moreprecisely,therelativeposi- tionbetweenpixels,meaningthewaypixelsarrangeinasegmentshouldaffectsegment length.AsshowninFig. 4.1 afactorof p 2isusedtocorrectthedistancebetweentwo pixelsindicatedbythe ! . 60 Figure4.1:Segmentlengthillustrationshowingtheneedforthefactor p 2forpixels joinedatedges. Theangleformedbysegmentsjoiningatajunctionisalsoanimportantfeatureof networklikeimage.Therearevariouswaystocalculatetheanglesformedatjunctions. Onewayistoplaceacircleofsmallradiusoforderseveralpixelsthatcentersatthejunc- tionandtoitsintersectionswithcracksegmentsasshowninFig. 4.2 .Theanglesof thearchescenteredatthejunctionandseparatedbytheintersectionswillbetheangles formedbysegments.TheothermethodforwhichaschematicpictureisFig. 4.3 usedin thisworkprovidesmoreaccuratevalues. Figure4.2:Onemethodforthejunctionanglebyusingintersectionpointsofthe crackpatternwithacircle. ThealgorithmofthemethodsketchedoutbyFig. 4.3 isthefollowing: a)Ateachjunction,traceallofitssurroundingsegments. b)Ineachsegment,foreachpixelwithinvicinity r s ofthejunction,calculatetheangles 61 Figure4.3:ThesamepixelpatternasFig. 4.2 withadifferentmethodforthe junctionangleswhichaverageanglesoverdifferentpixels. madeby Ox andthevectorconnectingitwiththejunction, a Ox c)Foreachsegment,averageovertheanglescalculatedinstepb). d)Repeatstepb)andc)forallsegmentssharingthejunction. e)Obtainallaveragedanglesofallofthesegmentsmadewiththehorizontalaxis,sort inascendingorderthenconvertthemtoanglesformedbetweensegments,similartoEq. ( 4.1 ). Thisalgorithmissimilarbutnotequivalenttocalculatingtheangularconnectivitymatrix atthejunctionpoint,inwhichtheeightpointconnectivitymatrixisreplacedbyacircle ofradius r s .Becausebesidescalculatingtheangle,thistaskrequireskeepingtrackof whichsegmentapixelbelongsto.Sinceeachsegmentcanbetrackedanditslengthcan becalculated,thecrackpatternnowcanbeflthresholdedflbytrimmingshortbranches. Thetrimmedcrackpatternhasonlylongbranchesremaining.Thispracticeishelpfulin somecasessuchaswhencalculatingcrackspacingbetweenwelldcracksegments. Forpatternsfoundinnature,suchastheveinleaforthepolygonalmudorlavacracks,it isfoundthat,initiallythejunctionsoccurrightangles.Later,asthepatternsevolve,the rightanglejunctionsarerelaxedinto120 o junctions,[ 36 ],[ 90 ]. Fig. 4.4 istheskeletonizedcrackpatternat6000 s inwhicheachsegmentiscoloredand thetotalnumberofpixelcomprisingitslengthisdisplayedinrandomcolors.InFig. 4.4 , 62 thesamecrackpatternisflcleanedupflbyremovingchildbrancheswhicharelessthana thresholdvalue(twovalueselevenandpixelsareusedtoshowtheofthe thresholdvalue. Figure4.4:Skeletonizedcrackpatternat6000 s .Eachsegmentiscoloredrandomly.The numbersplacedinthemiddleofeachsegmentrepresentthesegmenttotalnumberof pixels. Figure4.5:ThesameasFig. 4.4 exceptthatshortbrancheslessthan(left)andeleven (right)pixelsareremoved. Thenumberofloops,ortheirtotalarea,isanimportantquantitythatcharacterizesthe damagedegreeofthematerials.Inimageprocessing,thereisanalgorithmsforlabeling connectedregions,namedeflorflwavepropagationflprinciple,basedontherule thataflsweptoutflorflburnawayflpixelisnevertobevisitedagain,seeMathematical Morphology[ 200 ].Toidentifyloopinthecrackpatternimage,ofall,theMATLAB 63 Figure4.6:ZoomedinsectionofFig. 4.4 showsthevaluesoftheanglesateachjunction. function bwboundaries (level1)iscalledtoidentifyloopsandisolatedcracksalongwith theirexteriorboundaries.Thenanotherfunctioniscalledtodifferentiatebetweenaloop andanisolatedcrack.Itsalgorithmisthefollowing: (a)Pickarandompointintheinteriorboundaryoftheobjectwhichmightbeeitheraloop (aregioncomprisingofpixels1)oranisolatedcrack(asegmentcomprisingofpixels1) asdiscussedabove. (b)Determinewhethertherandompointisincontactwithanybackgroundpixel(0pixel). (c)If(b)istrue,marktheobjectasnotloop,exit.Otherwise,repeat(a)and(b)aslongas nomorethanrandompointshavebeentested.Ifmorethanrandompointshave beentested,marktheobjectasaloopthenexit. Thetotalareainpixelsofaloopedoranisolatedregioniscalculatedbycountingthetotal numberofpixelsthatuptheregionaftercallingtheMATLABfunction (level1). InFig. 4.7 ,eachisolatedcrackbranchislabeledasoneregionandcoloredinred.The boundariesoftheclosedregionsaremarkedinblueandeachbranchingcrack(child)is markedinthecyancolor.Childcracksstartfromajunctionpointofitsmothercrackasa resultofbifurcation(splitting)orkinking,asmentionedinSection 3.4 .Intermsofimage analysis,achildcrackisasegmentthatconnectsanendpointandthenearestjunction pixel.InFig. 4.8 ,thesamecrackpatternwithFig. 4.7 isplotted,inwhichthenumbers 64 displayedatthecentroidofeachloopindicatetheloopareasrescaledbythetotalsample area. Figure4.7:Skeletonizedcrackpatternat6000 s .Numbersdisplayedatthelowerright corneraretotalisolatedbranches,loops,childbranchesrespectively. Figure4.8:Crackpatternwithloopsat6000 s fromFig. 4.7 .Theareaofeachloopregion isrescaledbythedomainareaanditsvalueisdisplayedattheloopcentroid. Anothermorphologicalofimportanceinthisstudyisthecrackspacing.Thereare theoreticalstudiesofcrackspacingandpenetrationdepthofshrinkingslabsusingthe principleofenergyminimization,[ 39 ],[ 40 ],whichcanexplaintheperiodicdoublingin systemofparallelcracks.In[ 117 ],thespacingofthermalshockcrackingarraysisused toinverselyestimatetheconvectiveheattransfercoefCrackspacingisinterpreted 65 differentlydependingonthemorphologiesofthepatterns.Forexample,forthesystem ofparallelcracks,likecracksinquenchedsample,crackspacingismeasuredby theintersectionsofcrackswithastraightlineperpendiculartothem.Foranisotropic crackingnetwork,suchasamudcrack,orcolumnarcracks,thelinesplacedonthecrack patternmusthaverandomdirectionsandtheresultmustbeaveragedoverseveral differentlines.Inthisstudy,inthecaseofalocalizedheat(smallratioof l to L ), crackspropagateessentiallyradially.Thusitisnaturaltomeasurecrackspacingatdepth r c usinganarcofradius r c placedinthecrackpattern.Theaveragearcangle g across r = r c istakenasthearcangledividedbynumberofthearcintervals. Asanexample,considerthethirdplotfromthetop,ontheleft,ofFig. 4.10 forthedetailed stepsofdeterminingtheaveragecrackspacingat r = 0.5 H .First,theskeletonizedimage ofthecrackpatternisflcleanedupflbyremovingallbranchesshorterthanpixels. Intersectionpointsofthecrackswithanarcradius r c = 0.5 H arefoundbycomparing twosetsofbranchingpoints:(setA)isthesetofbranchingpointsofthecrackpattern alone,(setB)isthesetofthebranchingpointsoftheunionofthecrackpatternandthe arc.SetBisfoundbythedilationthenskeletonizationoftheunionimage.Becauseofthe pixelatednatureoftheimage,clusteredspuriousintersectionpointscanbegeneratedas showninFig. 4.9 .Intersectionpixelsthatarewithinacertaindistancearereducedtoone representativepoint.Hereafourpixeldistanceisusedasthecriterionforgrouping. Thisresultsintotalthirteenintersectionspoints N = 13asindicatedinrednumbers ofFig. 4.10 (bottomleft),andtheirpositionsareallHowever,forthecurrent purposeofcalculatingaveragecrackspacing,onlythepositionsoftheandlastpoints areneeded.Inparticular,inthiscase,thelocationsofthe1 st pointis ( x 1 , y 1 )=( 322,63 ) andofthe13 th oneis ( x N , y N )=( 678,67 ) .Thelocationofarccenteris ( x c , y c )=( 500,1 ) . Thearcangle g iscalculatedfromthefollowingtrigonometricrelation 66 Figure4.9:Spuriousintersectionpointsaregroupedtogether. g = arctan (( x c x 1 ) / r c )+ arctan (( x N x c ) / r c ) ,(4.2) whichyields g = 1.4573 ( rad ) .Theaveragecrackspacingisobtainedbydividingthearch angle g bythenumberofarcsegments N 1.Theresultshereis0.1214 rad or21.85 o . 4.2Morphologicalcharacterizationofnetwork-likepatterns Fracturepatternsbelongtoabroadergeometricalclass,thenetworkpattern,such asthoseofdendrite,Lichternbergtree,leafveins,insectandbirdnests,glaze,streams, rivers,streets,Internet,treebranches,mudcracks,soilcracks,etc.Thestudytheirmor- phologiescanrevealtheunderlyingphysicalandbiologicalrulesthatgeneratethesevar- iouspatterns.Theyalsohaveapplicationsinpatternsuchasbiologicaltax- onomy.Theavailableliteraturecontainsmorphologicalstudiesbasedontopoplogical grapththeory,[ 48 ],suchasthestudiesofrivernetworks,[ 41 ],[ 42 ]and[ 43 ],bonestruc- ture[ 50 ],insectnests[ 49 ]andantnetworks[ 47 ],[ 46 ]. Thenetworkpatternsareincludedinabroaderphenonmena,knownastheLipatterns, suchaswatercrystal,striationmarkonanimalshell,retiformcellsofinsectwings,ar- 67 Figure4.10:Intersectionsoftheflcleanedupflskeletonizedcrackpatternat6000 s withan arcofradius r c / H =0.0317,0.1583,0.2375,0.3167,0.4750,0.6333,0.7917,0.9500,fromleft toright,toptobottom,respectively.Rednumbersindicateintersectionpoints. rangementofcabbageleavesinsection,Irishmossseaweed,reptilianskinformations, cloudlikeformations,spiraldefects,angulatedforminerodedshale...Thestudyofpat- ternsfoundinnatureandtheirdynamics,knownastheLipatternstudy,hasbeenin existencefromancienttimeinChinesephilosophy.AsquotedfromGeorgeSteiner,(Life- lines)[ 201 ]: fiThereisahauntingifdeceptivemodernityinthenotion,sooftencelebratedbybaroquepoets andthinkers,thatarteriesandthebranchesoftree,thedancingmotionofthemicrocosmandthe solemnmeasuresofthespheres,themarkingonthebackofthetortoiseandtheveinedpatternson rocks,areallciphers.fl 68 Today,themorphologicalstudyofthepatternsandtheirdrivenforcesaswellasthe underlyingmechanismsbecomesincreasingpopular.Thecrackpatternsinthisstudy sharebothcharacteristicsofthetree-like(leafvein,Lichternberg,tree,fractal)andinter- connectedloop-like(street,neurite)patterns.Thetypeisdirectionalandtendsto forminsystemswithtransportationsuchasthatofnutrients,heatelectrical Thesecondtypeisisotropic,usuallyformedonasurfaceandfollowdirectlythe principalofenergyminimization,[ 52 ],[ 51 ],[ 47 ].Theycanbecharacterizedusingcrite- rionsuchascrackrank[ 53 ],spacing,length,junctionangle,numberofsides/vertexesfor eachloopdomain,loopdomaindirection,fractalnumbers[ 54 ],etc.Accordingto[ 54 ], theevolutionofaphysicalorbiologicalpatternoftenobeystheruleoflocallengthmini- mization.Furthermore,innature,therankofcrackscanbebasedoncrackwidthbecause oldercrackstendtobewiderthannewerones. Theimageanalysisalgorithmpresentedinthissectioncanbeappliedtoanypattern ingeneralwithnecessarypre-treatmenttoenhancethecontrastbetweenbackgroundma- terialsandcracksbeforeconvertingthemtoblackandwhiteimage.Somealgorithmsfor calculatingfractal(selfsimilarity)quantitiesusingimageanalysiswillbeoutlined.The programmingimplementationonthealgorithmshasnotbeendoneinthisthesis,butcan beextendedforfuturework.From[ 54 ],fractallacunarity,theparametercharacterizing theheterogeneityofafractalimage,isanimportantquantitythatisdiscussedinthis section.Assumeanimageof M totalpixelsisoverlaidbyboxesofsize L (pixels).The numberofboxessize L thatareneededtocovertheentireimage N ( L ) is N ( L )= K å m = 1 M m P ( m , L ) ,(4.3) inwhich P ( m , L ) isthepossibilitythattheboxsize L contains m pixelsand K isthemaxi- mumnumberofimagepointsthatfallinsidethebox.Foreachpixelintheimage,center itwithaboxofsize L ,thencountthenumberofimagepixels(pixel1)thatfallwithinthe box,namely F .Then,buildingahistogramof F inthefollowingway: P ( m , L ) isequalto 69 thenumberoftimesthat F attainsthevalue m ,dividedbytotalnumberofpixelscentered byboxeswhichisalsothetotalnumberofboxesconsidered.Thevariable log ( N ) / log ( L ) isasthefractaldimension,characterizingthespacenatureofthepattern. Afterobtainingthehistogramof F ,thefractallacunarityparameterisgivenby C ( L )= < m 2 > < m > 2 < m > 2 ,(4.4) whichisthevarianceoftherandomvariable F ,where <> istheaveragingoperator,and < m > and < m 2 > aretheaveragevaluesof F and F 2 ,inparticular < m > = K å m = 1 mP ( m , L ) ,(4.5) and < m 2 > = K å m = 1 m 2 P ( m , L ) .(4.6) Loopsinacrackpatternareandintermoftheirareaandtheir totalnumberintheprevioussection.Inthissection,morepropertiesofloopsaspolygons willbediscussed.Reference[ 129 ]presentsthealgorithmfornumberofvertices ofapolygonwhichcanbesummarizedasthefollowing: a)Foreachdomain,specifyitsboundary. b)Formasequenceofvectorswiththesamelengthinwhichtheheadofthenextvector isthetailofthepreviousvector. c)Calculatetheanglesformedbythesevectorsfoundin(b).Thelocalmaximaofthese anglesthevertexesofthedomain. Thisparameterisusefulfortheofpatternshavingmanyloopregions,such asmudcracksorglazesonceramic.Fig. 4.11 isanillustrationofthealgorithmabove,in whichthearrowsindicatesthemovementfromonepixeltothenext. 70 Figure4.11:Schematicalgorithmforvertexofapolygon.Eachpairofarrows correspondstoalocationofmaximumanglechange,whichindicatevertexes. Futhermore,eachlooporisolateddomaincanbecharacterizedbythefollowingshape factors: a)Areatoperimetersquaredratio(circularity), b)Aspectratio(ratioofFeret'sminimumlengthtomaximumlength), c)Convexity(ratiobetweenconvexhullperimetertoactualperimeter), d)Solidity(ratiobetweenareaofconvexhulltothedomainarea), e)Principaldirections(eigenvectors),eigenvalues. Forexample,following[ 129 ],thedomainprincipaldirectionscanbealsobasedonthe eigenvaluesandeigenvectorsofmomentofinertiatensor I givenby: I = 0 B @ I xx I xy I xy I yy 1 C A ,(4.7) inwhich: I xx = ZZ W ( x x c ) 2 dxdy , I yy = ZZ W ( y y c ) 2 dxdy ,(4.8) I xy = ZZ W ( y y c )( x x c ) dxdy , where ( x c , y c ) isthecentroidof W .Byusingthesummationruleofintegration,Eq.( 4.8 ) a , 71 indiscreteformreads I xx = N å i = 1 ( x i x c ) 2 , I yy = N å i = 1 ( y i y c ) 2 , I xy = N å i = 1 ( x i x c )( y i y c ) . Here N isthetotalnumberofpixelsinthedomainand ( x i , y i ) isthepixel'scoordinate, i = 1, N .Moreover,thecentroidistakenas: x c = N å i = 1 x i / N and y c = N å i = 1 y i / N ,which iscoordinateaveraging.Theprincipalmomentsofinertiaareeigenvaluesoftheinertia tensor,whichare I 1 = I xx + I yy 2 + s I xx I yy 2 2 + I 2 xy ,(4.9) and I 2 = I xx + I yy 2 s I xx I yy 2 2 + I 2 xy .(4.10) Thecorrespondingprincipledirection q p thecondition tan ( 2 q p )= 2 I xy I xx I yy .(4.11) Theratiooflargesttosmallesteigenvaluesisastheaspectratiowhichcharac- terizesthedomainelongation.Theshapefactorbasedonthearearatiobetweenthetrue objectandtheleastcircumscribedcirclecenteredontheobjectcenterofgravityisusedas acriteriontodistinguishingvoidsfrommicrocracks,see[ 130 ].Anotherwayofidentify- ingaregiondirectionispresentedin[ 138 ],inwhichanellipseisintotheisolated region.Themajorandminoraxesoftheellipseindicateregiondirections.Fig. 4.13 shows aloopthatisinscribedbyanellipse. 72 Figure4.12:Crackpatternwithloopsat6000 s fromFig. 4.7 .Atthecentroidofeachloop region,therearetwoarrowsindicatingitsinertialprincipledirections.Arrowlengthsare rescaledbyitsprinciplemomentofinertia:theratiooftwoarrowlengthsistheshape factor. Figure4.13:Inscribealoopwithanellipse,asshown. Accordingtothetheoryoforientationalordering,thedomainorderparametercanbe as S 1 = < cos 2 ( q ) > d 1 d 1 ,(4.12) inwhich q isthedirectionoftheloopdomainsinachosencoordinatesystem, d (equals twofor2D)isthespatialdimensionand <> istheaveragingoperator.Looplikepatterns canbeusingothermethodssuchasFourieranalysis.Inthismethod,thecrack densityinonedirectioniscalculatedbysummingallpixels(zeroforbackground,onefor 73 animage)intheotherdirection.Thenthisonedimensionaldensityistransformedintoa Fourierseries. Thecrackorientationandisotropycanalsobecharacterizedbytheintersectioncounting methodororientedsecantmethod,following[ 130 ].Asetofparallelequidistantlines titledatanangle q withthe Ox axisisplacedintothecrackpattern.Thenumberof intersectionpointsbetweenthecrackpatternandthelines N L ( q ) isafunctionofthe angle q asitvariesbetween0 o and180 o .Thedegreeoforientation w isgivenas w = N Lmax N Lmin N Lmax + p 2 1 ,(4.13) thussatisfying0 w 1,where N Lmax and N Lmin arethemaximumandminimum valuesofthefunction N L ( q ) .Ifthecrackpatternisisotropic,then N Lmax = N Lmin , whereby w = 0.Ifallcrackshavethesamedirection, N Lmin = 0,leaving w = 1. Inmaterialswithrandomlydistributedcracks,thedegreeofdamagecanbecharacter- izedbythecrackdensity.[ 139 ]derivestheformulasforcrackdensityfromthedamage variablewhichcanbeeitherscalarortensorial.Reference[ 136 ]calculatesthisparame- terfromfractureenergyandtheJ-integralforsolidmaterialsthatmayormaynothave stressinteractionwithTheresultsof[ 136 ]canbeusedforasystemofrandomly distributedslitcracksorabitraryconvexshapedcracks.Fortheformertypeofcracks, crackdensityisfoundtobegivenby d c = 8 M < l > 2 p 3 ,(4.14) where < l > isthemeancracklength, M = L A / l , L A = p N L /2isthetotalcracklength perunitarea,and N L isthenumberofcracksperunitlengthofthesample.Forthelatter one,itisequaltotheaveragevolumeofmicrovoidsperunitsubstancevolume,whichis e = N < r 3 a > ,(4.15) 74 where N isthenumbermicrocracksperunitvolumeand r a isthevoidradius.Reference [ 140 ]calculatesthechangesofmaterialpropertiessuchasYoung'smodulus,shearmod- ulusandPoisson'sratioforannealedandheavilycold-workedmetalsasafunctionofthe crackdensity. Relatedtocrackdensity,crackwidthisanothercharacteristicofcrackingpatternsof- tenconsideredinexaminingconcreteandsoftsubstanceonasubstrate.Crackwidthhas beenfoundtobedependentonsubstratefriction[ 141 ],samplethickness[ 56 ],[ 142 ],ma- terialcomponentsandloadingconditions.Fromtheresultsofnumericalsimulations,the crackwidthcanbeconstructedfromthedisplacementInexperimentalpractice,it canbetakenastheratiobetweencrackareaanditslengthintheskeletonizedimage. Intopologicalgeometry,aninterconnectedpatterncanbecharacterizedbytheMinkowski numbers.Reference[ 143 ]usesthreeMinkowskinumberstoquantifythecrackmorphol- ogyinadryingsoil,namely, M 0 , M 1 , M 2 ,inwhich M 0 isthecracksurfacearea, M 1 isthe totalcracklengthand M 2 istheEulernumber.Inimageanalysispractice,the M 0 valuein pixelsquaredunitsisequivalenttothetotalnumberofblackpixelbeforeskeletonization. M 1 isthelineintegralalongthecrackpath: M 1 = Z L ds ,(4.16) whichyieldsthecracklength.Asdiscussedintheprevioussection,Section 4.2 ,when neglectingtherelativearrangementofpixelsinacrackpath,thelineelement ds equals onepixel.Therefore M 1 canbetakenasthetotalnumberofblackpixelsintheimageafter skeletonizing.Otherwise, ds shouldbecorrectedbya p 2factorinthediscreteformof theintegralusingsummationrule.Thelastnumberisas: M 2 = Z L ds r (4.17) where r istheradiusofcurvaturealongthecrackpath.Evaluatingtheaboveintegral onaboundarygives2 p foraclosedconvexboundary(correspondingtoanobject)and 75 2 p foraclosedconcaveboundary(alooporhole).Therefore M 2 equalsthenumber ofobjectsminusthenumberofloopsandisusedasatopologicalnumberthatdescribes connectivity. 4.3ofcrackmorphology Inthissection,thecrackingpatternscorrespondingtoonesetofparametersthatare producedinSection 3.4 areusingtheimageanalysisalgorithmpresentedin theprevioussections.Themorphologyofthecrackingpatternchangesastimeprogresses throughtheevolutionstagesasoutlinedearlier:(a)inertheating,notpyrolysis;(b) pyrolysis,crackinitiation,notcharred;(c)partiallycharred,mainlyinitiation,(d)crack elongationandfastpropagation;(e)almosttotallycharred,slowpropagationtostop. Amongthesestages,crackmorphologyiswellinlatestage(d)andearlystage(e) andhaslittlechangeasthesimulationproceeds. Fromasequenceofmaximumprinciplestressplots,itcanbeseenthatatthelatestage (d),cracksformedpreviouslyareactivelypropagatingandsplittingandnucleationhas notyetdiminished.Wheneachevent,aspropagation,splittingornucleationhappens, thenumberofcracks,cracktipsandjunctionschanges.Inelongation,thenumberofeach kindremainsthesame.Withinitiation,onenewcrackisformedalongwithtwocrack tips.Oneofthetipremainsinactivewhiletheotherpropagatesintothecharringdomain, formingalongercracksegment. Ajunction,thecommonpointbetweenthreecracksegments,canbeformedviadifferent mechanisms.Oneisbifurcation,thesplittingofonemothercrackintotwochildcracks therebyturningthemothercracktipintoajunction.Henceeachtypeincreasesbyonein number.Ajunctionisalsoformedwhenachildcrackspringsofffromitsmothercrack, usuallyatakinkpointatwhichstressesareraisedbythesharpturn.Anothereventthat createsanewjunctioniscrackintersection,thesocalledT-junction,becauseitoccursata rightangle.Oneintersectiontradesonetipforonejunction,decreasingthetotalnumber 76 ofcracktipswhileincreasingthetotalnumberofjunctionbyoneforeach. Inaskeletonizedcrackimage,theendpointandjunctionpixelsareandcounted ateachtimestep,forexampleat6000 s asseeninFig. 4.14 .Theevolutionoftheirtotal numbersareshowninFig. 4.15 .Fromthee,itcanbeseenthat,attheinitiationstage (thethirdstage),thenumberoftotalcracktips(cross)increasesatafasterratecompared withtotaljunctions(dot)duetonewlycreatedcracks.Thusthedifferencebetweentotal numberofcracktipsandjunctionsincreasesatthethirdstage.Itthenremainsconstant duringtheevolutionandpropagationstages(thethirdandearlyfourthstages)because elongationactivitypredominates.Afterthat,attheslowpropagationandloopstage(late fourthandstages),thedifferencedecreasesduetoprevailingintersection. Figure4.14:Skeletonizedcrackpatternat6000 s .Cracktipsaremarkedbytheletterein blueandjunctionsaremarkedbytheletterbinred.Whenonecrackintersectsanother, itstipisreplacedbyanintersectionpoint. Thestatisticsofsegmentlengthandjunctionanglesarecalculatedforthewelldeveloped crackpatternofSection 3.4 .6000 s ischosenbecauseatthistimecrackpatternhasalready evolvedthroughallstages.Thealgorithmsforcalculatingsegmentlengthandjunction anglearediscussedinthepreviousSection 4.1 .Theyareappliedforthewholeframe, calculatingthelengthofeachsegmentandanglesofeachjunction.Fig. 4.16 showsthe probabilitydistributionofsegmentlengthwhichisrescaledbythelengthofheatedregion l .Asseenfromthee,theshortersegmentshavehigherfrequencyofoccurrenceand 77 Figure4.15:Theevolutionoftotalnumberofcrackjunctions,cracktipsaswellastheir difference.Thedifferencetrendsareincreasing,remainingconstantanddecreasingfrom thethirdstagetothestage. viceversa.ThePDFofthesegmentlengthshowexponentialdecayingbehaviorwhichis acharacteristicoffractalorself-similarity,see[ 131 ],[ 133 ],[ 134 ].Moreover,inanumerical studyoftectonicrupture[ 132 ],thedistributionandfractalparametersoffracturelength aredeterminedbydifferentphasesandmechanismsofthefaultingprocess,suchasnu- cleation,growthandcoalescence. Figure4.16:Probabilitydistributionfunctionofsegmentlengthcorrespondingtocrack patternat6000 s inFig. 4.4 . 78 Theprobabilitydistributionofthejunctionanglescorrespondingtothecrackpatternat 6000 s isshownbyFig. 4.17 .Thejunctionanglehasmeanvalue m a = 119.6803 o with standarddeviation s a = 40.2904 o .Theslightdeviationof m a from120 o impliesthatmost ofthejunctionsaretriple.Theangleofhighestfrequencyisaroundthismeanvalue(Y- junction)andtheoneofthesecondpeakisinproximityto90 o (T-junction). Figure4.17:Probabilitydistributionofjunctionanglescorrespondingtocrackpatternat 6000 s inFig. 4.4 Someofveryshortsegmentsarenottruecracks,butareresultsfromtheprocessof thinningtheimage.Thereforethestatisticsofflcleanedupflcrackpatternisalsoper- formedandcomparedtothoseoftheoriginalpatternabove.Fig. 4.18 and 4.19 showsthe distributionofsegmentlengthandjunctionanglecorrespondingtothethresholdedcrack patternat6000 s inwhichsegmentsoflessthanpixelsareremoved.Thedensityof theshortersegmentsdecreaseasadirectconsequenceofthetrimmingprocess. InFig. 4.20 ,theaveragecrackspacing g ofthecrackpatterninSection 3.4 at6000 s as afunctionofcrackdepth r c isplotted,inwhich r c isnormalizedbythelengthofheated surface l .Fromthee,itcanbeseenthattheoveralltrendisadecreaseinspacing exceptfor r c > 3.5 l .Ontheotherhand,crackintersectionincreasesthespacingbetween 79 Figure4.18:ProbabilitydistributionfunctionofsegmentlengthcorrespondingtoFig. 4.5 whenbranchesarethresholded. Figure4.19:ProbabilitydistributionofjunctionanglescorrespondingtoFig. 4.5 when branchesarethresholded. branches,resultinginlocalfljumpsflofthecurve.Closerto r c / l = 1,crackspacingde- creasesexponentiallyduetobranchingandnucleation.From r c / l = 1to r c / l = 3.5,crack spacingremainsconstantduetoanapparentbalancebetweenbranchingandintersection .Afterthat,branchingdiminishesandtheaveragecrackspacinggrows. AsmentionedinSection 3.4 ,crackspacingcorrelateswiththedensitygradient.Inpar- ticular,closetotheinsulatedlowersurfacewhere d islarge,thedensitygradientissmall, 80 Figure4.20:Theaveragecrackspacingasafunctionofnormalizeddepthradius r c / l whenthecrackpatternat6000 s isthresholdedusingpixels. andthecracksarespacedmorewidelyapart.Thereverseappliestotheotherregionsthat areawayfromthelowersurface.Thisechoesaof[ 38 ]madeinastudyofcolum- narcracksformedbytemperaturegradientsinrocks.Itisseenthattheoveralltrendof thecrackpatternisadecreaseinspacinguntil r c > 3.5 l .Inthise,notethatwith intersection,thespacingstillremainsquiteconstantduetoabalancebetweenbranching andintersection.Fewerbranchescangrowintotheregion r c > 3.5 l .Accordingly,the branchingdiminishesandthecrackspacinggrows. 81 CHAPTER5 DIMENSIONLESSGROUPSANDMATERIALPROPERTIES 5.1ScalingAnalysis Thepurposeofthischapteristothedimensionlessgroupsthatcouldcharacterize thescalingoftheplanestressinitialboundaryvalueprobleminSection 2.2 and 2.3 .The variablesandtheirunitsintheproblemaresummarizedinthetable 5.1 below. Fromthetable,itcanbeseenthattheheatconduction-pyrolysis-elasticityproblemcon- tainssixteenconstants,inwhichthePoisson'sratio n andthemasslosscoef g are dimensionlessandtheotherfourteenhaveunitscomposedfromfourbasicunits.Ofthese otherfourteen, q 0 and k y onlyappearsasaratioandsothisleavesthirteenconstantswith units: L , H , a x , a y , T 0 , q 0 k y , l , r c , A , T a , r 0 , E , s c .Thefourbasicunitscanbetakenaseither length/time/temperature/energyorlength/time/temperature/mass.Byusingthe BuckinghamPitheorem,thethirteenprecedingdimensionalconstantscombinetoform ninedimensionlessPigroups. Materialscouldbedividedintothermallythinorthermallythickdependingonthera- tioofheatpenetrationdepththatisafunctionofheatthermaldiffusivityandmate- rialthickness.Theheatpenetrationdepthcanbenedintermsofcharacteristiclength forheat l f = k y T 0 / q 0 .Thermallythicksamplehastheratio l f / H or k y T 0 / ( Hq 0 ) beasmallvalueandviceversa,thevalueislargeforthermallythinsample.Forboth thermallythinandthermallythickcase,theignitiontime,hasbeenderivedanalytically in[ 144 ],whichshowsdependenceonheatthermaldiffusivity,initialandignition temperature.Accordingto[ 144 ],whenneglectingradiationheatloss,forthermallythick, theignitiontimegrowsas q 2 0 whileforthermallythin,itgrowslike q 1 0 . ThechoiceofthecharacteristictimeofheatconductionasusedinSection 3.3 is moreformallyForthispurposeregardsuchatime t hc asstilltobedetermined. 82 Materialunits2Dunits2DEquationof constantsintermsofenergyintermsofmassappearance L domainlength lengthlength H domainwidth lengthlength a x thermaldiffusivity length 2 time length 2 time ( 2.10 ) inxdirection a y thermaldiffusivity length 2 time length 2 time ( 2.10 ) inydirection k y thermalconductivity energy time temperature mass length 2 time 3 temperature ( 2.29 ) inydirection T 0 initialtemperature temperaturetemperature ( 2.25 ) q 0 heatparameter energy time length mass length time 3 ( 2.30 ) l lengthofheatedregion lengthlength ( 2.30 ) r c limitingsoliddensity energy time 2 length 4 mass length 2 ( 2.2 ) A preexponentfactor time 1 time 1 ( 2.2 ) T a activationtemperature temperaturetemperature ( 2.2 ) r 0 initialsoliddensity energy time 2 length 4 mass length 2 ( 2.2 ) EYoung'smodulus energy length 2 mass time 2 ( 2.12 ) n Poisson'sratio ( 2.12 ) g masslosscoef ( 2.12 ) s c tensilestrength energy length 2 mass time 2 ( 2.9 ) FieldunitsunitsEquationof variablesintermsofenergyintermsofmassappearance Tsolidtemperature temperaturetemperature ( 2.10 ) r soliddensity energy time 2 length 4 mass length 2 ( 2.2 ) # straintensor ( 2.12 ) s stresstensor energy length 2 mass time 2 ( 2.12 ) udisplacement lengthlength inthehorizontaldirection vdisplacement lengthlength intheverticaldirection Table5.1:VariablesandtheirunitsoftheprobleminSection 2.2 withinitialandboundary conditions 2.3 . 83 Inaddition,let t = t / t hc , z = x / H , h = y / H (thus0 < z < L / H ,0 < h < 1)and q =( T T 0 ) / T 0 . ThedimensionlessformoftheenergyequationEq.( 2.10 )is ¶q ¶t = t hc a y H 2 a x a y ¶ 2 q ¶z 2 + ¶ 2 q ¶h 2 ! ,(5.1) thusthetimescale t hc whichcharacterizesEq.( 2.10 )thatnaturallyemergesis t hc = H 2 / a y ,(5.2) whereuponEq.( 5.1 )becomes ¶q ¶t = a x a y ¶ 2 q ¶z 2 + ¶ 2 q ¶h 2 ! .(5.3) Notethat t hc asgivenbyEq.( 5.2 )coincideswiththatusedinSection 3.3 .Theheat q 0 canalsobeusedtocharacterizetheheattime t flux byusingEq.( 2.29 ),( 2.30 ).In particular,thedimensionlessformofEq.( 2.29 )usingEq.( 2.30 )is ¶q ¶h = q 0 H k y T 0 (5.4) FromEq.( 5.2 ), H = t hc a y H andsubstitutingthisintoEq.( 5.5 )yields ¶q ¶h = q 0 t hc a y Hk y T 0 .(5.5) whichcanbeinterpretedastheratiooftwocharacteristictimes,oneofthemis t hc and theother,usingEq.( 2.30 ),canbeas t flux = Hk y T 0 q 0 a y ,(5.6) thus( 5.5 )becomes ¶q ¶h = t hc t flux and t flux by( 5.6 )characterizestherelationof thermalboundarycondition( 2.29 )totheheatconductionequationEq.( 2.10 ). FromEq.( 2.2 ),let r = r / r 0 ,thus 84 ¶ r ¶t = t hc A ( r r c ) e T a / ( T 0 ( q + 1 )) ,(5.7) whichisequivalentto ¶ r ¶t = t hc Ae T a / T 0 ( r r c ) e T a T 0 q q + 1 ,(5.8) thusthechemicalreactiontimescalethatcharacterizesthepyrolysisreactionasdescribed byEq.( 2.2 )couldbetakenas: t chem =[ A exp ( T a / T 0 )] 1 and( 5.9 )becomes ¶ r ¶t = t hc t flux ( r r c ) e T a T 0 q q + 1 ,(5.9) TwoPigroupscanbechosenasthelengthratios, P 1 = H / L and P 2 = l / L ,whichcharac- terizetheproblemgeometry.ThethirdPigroup P 3 = a y / a x characterizestheanisotropy oftheheatconductionproblem,itcanalsobeviewedasbeingderivedfromthedimen- sionlessEq.( 5.3 ).Twoothergroupsarechosenastheratioofthetwocharacteristictimes: P 4 equals t hc / t flux ,or q 0 H / ( k y T 0 ) and P 6 equals t hc / t chem or H 2 Ae T a T 0 / a y . P 4 isalso equalto l f / H whichcanbeusedtodistinguishedthermallythickfromthermallythin problemasmentionedabove.Thegroupischosenastheratiooftwotemperatures P 5 = T a / T 0 ,alongwith P 6 ,relatingthethermalandthepyrolysisprocesses.Thesev- enthgroupisastheratiooftwodensities P 7 = r c / r 0 ,gtheextentof pyrolysis. ThelasttwogroupsmustcontainYoung'smodulus E andtensilestrength s c asthey havenotappearedinthesevengroupsfrom P 1 to P 7 yet.Bothofthetwoparameters havethesameunit energy / length 2 .Letgroup P 8 have E ,thus P 8 mustalsoincludeone oftheotherparameterswhichhave energy intheirunits,more k y , q 0 , r c , r 0 . Thesameargumentfollowsifonechooses mass insteadof energy asthebasicunit.Since Young'smodulusisassociatedwithstress(forceperunitarea)anddensityisaninertial term,theirgroupingismorereasonable.Theratio E / r 0 hasunit time 2 / length 2 which isrelevanttosomecharacteristicvelocity.Therearedifferentwaystochose P 8 thatis 85 acombinationof E / r 0 ,uponconsiderationthatthepyrolysisproblemprovidesdirect inputforthestressproblem, P 8 ischosenas E / ( r 0 ( AH ) 2 ) . Sinceboth E and s c havethesameunit,itisnaturaltolet P 9 = s c / E ,theratioof thecrackingstresstotheelasticmodulus,asthedimensionlessgroup. P 9 isa crackresistanceparameter,whenitissuflarge,nocrackisabletoform.The offracturemechanicsoftenseparatestwodistinctprocess,crackinitiationandcrack propagation.Thetheoryofthermoelasticitysuggeststhatmaterialshavinghightensile strength,thermaldiffusivity,lowYoung'smodulusandundergoslowthermalexpansion [ 135 ]duringheatingcanhavebettercrackresistance.Duringcooling,mostmaterials contract,orshrink,justasthepyrolyzingsolidinthisstudycontractswhenitlosesmass. Thus,thethermalcontractioncoefincoolingisanalogoustothecurrentmassloss coefbecausebothserveascoefoftheshrinkagestress. Ourproblem,whichischaracterizedbyninePigroups,willemploycharacteristic unitsfromtheparameters H , t hc , T 0 and r 0 (length,time,massandtemperature).From Eq.( 5.3 ),Eq.( 2.10 )canbewrittenas: ¶q / ¶t = P 3 ¶ 2 q / ¶z 2 + ¶ 2 q / ¶h 2 (5.10) Theinitialconditionforthedimensionlesstemperature q is q = 0,whichisalsothe boundaryconditionatthetwolateralsides, z = 0, L / H .Theboundaryconditionfor q ontheinsulatedside( h = 0)is: ¶q / ¶h = 0,andontheheatedside( h = 1),Eq.( 2.29 ) andEq.( 2.30 )give ¶q ¶h = 8 > > < > > : 0if j z 1 2 P 1 j > P 2 2 P 1 P 4 if j z 1 2 P 1 j P 2 2 P 1 . (5.11) ThepyrolysisequationEq.( 2.2 )or( 5.9 )becomes ¶ r / ¶t = ( r P 7 ) 1 P 6 e P 5 q 1 + q .(5.12) Finally,thestressesarenon-dimensionlizedwithrespecttoYoung'smodulusEusing 86 E =( P 8 / P 2 6 ) exp ( 2 P 5 )[ r 0 H 2 / t 2 hc ] , viz. s xx = s xx / E , s yy = s yy / E , s xy = s xy / E , s zz = s zz / E , s p = s p / E ,where s p isthe relevantprincipalstress. Whenconvectiveorradiativeheatcoolingisincludedinthemodelviatheheat terminEq.( 2.29 ),thereexistsanothertemperature T m whichisthemaximumtemper- atureobtainedsincetheincomingheatisbalancedbyheatlossterms.Thenthe Arrheniusequationcouldberescaledinthefollowingway,usingdifferentdimensionless temperature q =( T T 0 ) / ( T m T 0 ) whichvariesbetween0and1,Eq.( 2.2 )isrewritten as: ¶ r ¶t =( r r c ) At 0 e T a T 0 e T a 1 T 1 T 0 .(5.13) Considertheterm e T a T whichcanberewrittenas e T a T = e T a T m e T a T T a T m = e T a T m e T a T m T T m T = e T a T m e T a T m T m ( T m T 0 ) q T 0 T 0 +( T m T 0 ) q + T m T m = e T a T m e T a T m ( T m T 0 ) 1 q T m ( T m T 0 )( 1 q ) = e T a T m e T a T 2 m ( T m T 0 ) 1 q 1 s ( 1 q ) . inwhichthefollowingdimensionlessquantitiesareused: b = T a T 2 m ( T m T 0 ) , s = 1 T 0 T m , D a = At 0 e T a T m ,thentheArrheniusEq.( 2.2 )writtenindimensionlessformis ¶ r ¶t = D a ( r r c ) exp b 1 q 1 s ( 1 q ) ,(5.14) 87 ascomparedwith( 5.9 ). Furthermore,inthisproblem,thestressbalanceistakentobequasisteady,containingno inertiaterm.Ifinertialeffectistakenintoaccount,thestressbalancein x directionwith stressstraindisplacementrelationsfollowingEquations( 2.19 ),( 2.19 ),( 2.19 )and( 2.22 )is: r 0 ¶ 2 u ¶ t 2 = E ¶ 2 u ¶ x 2 + s . t . s + g 1 2 n 1 r 0 ¶r ¶ x ! ,(5.15) where s . t . s aresometermssimilarto ¶ 2 u ¶ x 2 ,thesecondorderspatialpartialderivativeof displacement,suchas ¶ 2 u ¶ y 2 , ¶ 2 u ¶ x ¶ y ,etc.Thedimensionlessformof( 5.15 )is ¶ 2 u ¶t 2 = Et 2 hc r 0 H 2 ¶ 2 u ¶z 2 + s . t . s ! + Et 2 hc r 0 H 2 g 1 2 n ¶ r ¶z .(5.16) where u = u / H .Thus,inthiscase, P 8 canbechosenas Et 2 hc r 0 H 2 andanothercharacteristic timethatscalesthewavepropagationinsidesolidistakenas: t wave = p r 0 H p E . Insummary,fromthissection,theeightdimensionlessgroupscharacterizingthethermal- pyrolysis-mechanicalproblemaredeterminedasthefollowing 88 P 1 = H L P 2 = l L P 3 = a y a x P 4 = q 0 H k y T 0 P 5 = T a T 0 (5.17) P 6 = H 2 Ae T a T 0 a y P 7 = r c r 0 P 8 = E r 0 ( AH ) 2 P 9 = s c E 5.2MaterialProperties Thissectionwillreviewthethermalandmechanicalpropertiesoffourclassesofim- portantmaterials(cellulosic,thermoset,thermoplastic,XGnP)ofspecialinterest. Cellulosicmaterials,whicharestudiedextensivelyinliterature,areimportantchar- ringpyrolysissubstance.Existinginnature,cellulose,alongwithotherlongchainednat- uralpolymerssuchhemicellulose,lignin,magnan,xylan,arefoundinwood.Sincewood haslongarrangedinacylindricalstructure,theirthermalandmechanicalproperties ofwoodisorthotropic.Thevaluesofthesequantitiesalsovarydependingontypesof wood.TheYoungsmodulus E andPoissonsratio n ofvariouskindsofwoodspecies,tak- ingintoaccountitsorthotropy,couldbefoundintable4.1and4.2of[ 22 ]respectively.Itis notuncommontopropertiesforwoodtobetabulatedasifthewoodisisotropic.In cylindricalcoordinate,let z thedirection, r theradialdirectionand q thetangential 89 direction.AmongsixPoissonsratio, n r q oftenhashighestvalueand n z q haslowestvalue. Woodtypicallyhaslowthermalconductivity(lessthan1 Wm 1 1 K 1 ),whichmakesita goodchoiceforthermalinsulation. [ 156 ]considerscellulosepyrolyzingviaonesinglestepreactionormultipleonestep chemicalreactions(celluloseto C 2 H 4 , CH 4 , C 2 H 6 , C 3 H 6 , CH 3 OH , CH 3 CHO , H 2 O , CO , CO 2 ,etc)viaEq.( 5.18 )andprovideskineticsparametersforeachreactionwhichare tabulatedintableIIof[ 156 ].Table4of[ 167 ]provideskineticsparametersforAlcell andKraftligninatdifferenttemperatureranges.Theligninpyrolysismodelin[ 169 ] issimilartothemodelin[ 156 ]forcellulose,inwhichlignindecomposesinto CO , CH 4 , CO 2 , C 2 H 4 , C 2 H 6 , H 2 O , HCHO , H 2 O + HCHO , C 3 H 6 andchar.Othermodelsincluding [ 168 ]accumulateallvolatilesi th intoasinglevolatileterm. dV i dt =( V , i V i ) A i e E a , i RT ,(5.18) where V i isvolumeofgasspeciesi th , V , i istheavailablevolumeofgasspeciesi th , A i and E a , i arefrequencyorpre-exponentfactorandactivationenergyofthereactionfor creationofgasspeciesi th .Whenallgasspeciestermsareaccumulatedintoasinglegas term,thekineticsparametersareequaltothoseofEq.( 2.2 )forsoliddecomposition.From [ 160 ],theactivationenergyforthedevolatilizationofcellulose,hemicellulose,ligininare 236 kcal / mole ,100 kcal / mole ,46 kcal / mole respectively.Table 5.2 belowsummariesthe studyofcellulosicpropertiesfoundintheliterature. 90 PropertiesCellulosicmaterials a z 10 6 ( m 2 s ) [ 161 ] birch 0.307to0.513 a r q 10 6 ( m 2 s ) [ 161 ] birch 0.335to0.537 k z ( J mKs ) [ 161 ] birch 0.291to0.37 k r q ( J mKs ) [ 161 ] birch 0.177to0.25 k ( J mKs ) [ 170 ] pinewood 0.166 + 0.396 X r c [ 169 ] milledwoodlignin 14% r 0 ,[ 160 ] hardwood,charyield 14 23%, soft wood,charyield 20 26% r 0 ( kg m 3 ) [ 162 ] whiteoak 842.0, basewood 461.0,[ 170 ] pinewood 590-640 T a ( K ) A ( 1 s ) [ 146 ] 2.1 10 8 ,[ 155 ] cellulose 1.7 10 4 ,[ 23 ] cellulose 1.2 10 6 ,[ 146 ] 2.1 10 8 , [ 149 ] cellulose 0.019-0.14,[ 151 ] beechsawdust ( T 600 K ) ,0.0053,[ 151 ] T 600 K ,2.3 10 4 ,[ 151 ] cellulose 6.79 10 3 ,[ 151 ] cellulose 6.79 10 9 , [ 152 ] cellulose 3.9 10 11 ,[ 170 ] cellulose 0.7 10 5 ,[ 153 ] lignin 1.2 10 8 , [ 153 ] hemicellulose 1.45 10 9 ,[ 154 ] wood 10 8 [ 171 ] cellulose 6.8 10 9 ,[ 168 ] lignin 78.3 ( 160 K 680 K ) ,1.5 ( 410 K 1680 K ) ,[ 169 ] milledwoodlignin 5650or10 5.53 [ 163 ] lignin 1.65 10 4 ( 280 K 344 K ) ,0.0933 ( 344 K 435 K ) ,[ 164 ] lignin 7.16 10 10 ( 280 K 300 K ) ,[ 165 ] periodatelignin ( 260 K 375 K ) [ 166 ] Aspenwoodlignin 4.43 10 6 [ 150 ] cellulose,innitrogen 6.06 10 9 , n = 0.46, cellulose,insteam 1.67 10 9 , n = 0.51,[ 170 ] hemicellulose 2 10 10 Table5.2:Literaturestudyofcellulosicmaterialproperties. 91 Table 5.2 (cont'd) [ 172 ] Pinuspinasterwood,sevencomponents,component 4.2 10 6 , sec- ondcomponent 1.2 10 11 ,[ 172 ] Pinuspinasterbark,sevencomponents, component 3.3 10 4 , secondcomponent 2.2 10 12 ,[ 172 ] Chestnut treewood,eightcomponents,component 1.9 10 10 , secondcomponent 3.0 10 15 ,[ 172 ] Cellulose-Whatmanpaper 6.8 10 12 [ 173 ] almondshell 1.37 10 6 , hazenutshell 1.34 10 6 , beechwood 8.09 10 10 E a ( kJ mol ) [ 155 ] cellulose 79.4,[ 23 ] cellulose 100.5,[ 146 ] 101 + 142 X , [ 149 ] cellulose 8.8-33.4,[ 151 ] beechsawdust ( T 600 K ) ,18,[ 151 ] ( T 600 K ) ,84,[ 151 ] cellulose 71,[ 151 ] cellulose 139,[ 152 ] cellulose 166.4,[ 170 ] cellulose 146,[ 153 ] lignin 141.3,[ 153 ] hemicellulose 123.7,[ 154 ] wood 125.4 [ 171 ] cellulose 33.4,[ 168 ] lignin 25.08 ( 160 K 680 K ) ,30.54 ( 410 K 1680 K ) ,[ 169 ] milledwoodlignin 81.2 [ 163 ] lignin 86.52 ( 280 K 344 K ) ,37.08 ( 344 K 435 K ) ,[ 164 ] lignin 143.3 ( 280 K 300 K ) ,[ 165 ] periodatelignin 35.9 75.7 ( 260 K 375 K ) [ 166 ] Aspenwoodlignin 57.5 168.92,[ 160 ] lignin 46 [ 150 ] cellulose,innitrogen 153.13, n = 0.46, cellulose,insteam 143.09, n = 0.51,[ 160 ] cellulose 236 [ 160 ] hemicellulose 100,[ 174 ] hemicellulose 110,[ 170 ] hemicellulose 83 [ 172 ] Pinuspinasterwood,sevencomponents,component 83, second component 146,[ 172 ] Pinuspinasterbark,sevencomponents,compo- nent 52, secondcomponent 159,[ 172 ] Chestnuttreewood,eightcomponents, component 117, secondcomponent 188,[ 172 ] Cellulose-Whatman paper 167 [ 173 ] almondshell 92.82, hazenutshell 92.36, beechwood 123 ˘ 150 92 Table 5.2 (cont'd) t chem 1 A e E a RT 0 ! [ 155 ] cellulose 3.93 10 9 ,[ 23 ] cellulose 1.59 10 12 , [ 149 ] cellulose 1792.7 4.67 10 6 ,[ 151 ] beechsawdust ( T 600 K ) ,2.56 10 5 , ( T 600 K ) ,1.83 10 10 [ 151 ] cellulose 3.39 10 8 ,2.34 10 14 ,[ 152 ] cellulose 2.41 10 17 [ 153 ] lignin 3.34 10 16 ,[ 153 ] hemicellulose 2.38 10 12 [ 154 ] wood 6.837 10 13 ,[ 171 ] cellulose 9.6 10 5 [ 168 ] lignin 297.3,[ 164 ] lignin 1.24 10 14 [ 166 ] Aspenwoodlignin 2320.5 E ( GPa ) [ 162 ] whiteoak 18.4, basewood 14.0 s c ( MPa ) n g c p ( J kgK ) [ 172 ] pinewood 1950,[ 172 ] charresidue 1350 P 9 ( s c / E ) P 3 ( a x / a y ) ˘ 1.0(neglectanisotropy) P 7 ( r c / r 0 ) ˘ 0.2 P 8 H 2 E r 0 A 2 t hc H 2 ( a 1 y ) ˘ 3 10 6 t flux q 0 HT 0 k y a y ˘ 5 10 5 Besidecellulosic,otherimportantmaterialsarerubber,thermoset,thermoplastics, XGnPanditscomposites.Rubberiscomposedofelastomerswhichareelasticandhave crosslinkedchains.Itcanbederivedfromnaturalrubbertreeorfromcrudeoil.Ther- mosetandthermoplasticsaretwokindsofplasticsubstancesinwhichthetypere- 93 mainsforeverinsolidstateonceitiscured[ 175 ].xGnPisacarbonderivednanomaterial thatisnewlydevelopedwhichhasexcellentphysicalproperties,suchaslowweight,high stiffnessandtensilestrength,veryhighthermalconductivity,[ 182 ],[ 184 ],[ 185 ].xGnP nanoplatelets,whenarrangedhorizontallyinmaterialsexposedtocandissipate heattothelateralsidesandserveasgasandthermalbarrierbetweentheheatand unburnedsubstance.Theirmaterialspropertiesarealllistedinthefollowingtables, 5.3 , 5.4 , 5.5 , 5.6 . 94 PropertiesRubber a ( m 2 s ) [ 177 ] k r 0 c p = 4 10 8 ,Figure7of[ 176 ] conductivenaturalrubber(40 HAF/NR)B2C 0 50% at 70 o C 10 10 8 18 10 8 k ( J mKs ) [ 177 ]0.11,[ 179 ],Figure7of[ 176 ] conductivenaturalrubber(40HAF/NR) B2C 0 50% at 70 o C 2.1-4.5, CNT/SBRBR 0.08-0.13 r c ( kg m 3 ) [ 178 ]400.0 r 0 ( kg m 3 ) [ 177 ]1200.0,[ 179 ] (SBR) 956.0,[ 179 ] (NBR) 935.0 [ 179 ] (IIR) 925.0, (CR) 1275, (NR) 914.0 T a ( K ) A ( 1 s ) [ 177 ]2 10 9 ,[ 180 ] (nitrile) 10 12 ,[ 181 ] (tyre) 10 10 Fluoroelastomers 5.2 10 18 E a ( kJ mol ) [ 177 ]76 [ 180 ] (nitrile) 101.0,[ 180 ] (neoprene) 96.3,[ 180 ] (natural) 113.0 [ 180 ] (EPDM) 92.0,[ 180 ] (PVC/nitrile) 105.0 Fluoroelastomers 210.8 t chem 1 A e E a 2 RT 0 ! [ 177 ]0.00206,[ 180 ] (nitrile) 0.000621, Fluoroelastomers 0.368 t chem 1 A e E a RT 0 ! [ 177 ]8555.06,[ 180 ] (nitrile) 385742.3, Fluoroelastomers 7.07 10 17 E ( GPa ) ethylenepropylenerubber 1.21, rubber 0.01 naturalrubber 1.2 s c ( MPa ) [ 180 ] (nitrile) 16.0,[ 180 ] (neoprene) 17.8,[ 180 ] (PVC/nitrile) 14.6 naturalrubber 14.2 n [ 177 ]0.49 Table5.3:Propertiesparametersofrubber. 95 Table 5.3 (cont'd) c p ( J kgK ) [ 177 ]2200.0,[ 179 ],Figure7of[ 176 ] conductivenaturalrubber(40 HAF/NR)B2C 0 50% at 70 o C 15000 35000 P 9 ( s c / E ) naturalrubber 0.0118 P 3 ( a x / a y ) 1.0(rubberisisotropic) P 7 ( r c / r 0 ) [ 177 ] tirepyrolyzedininertenvironment 33 38% P 8 H 2 E r 0 A 2 2.5 10 16 10 21 t hc H 2 ( a 1 y ) [ 177 ]25 10 6 t flux q 0 HT 0 ( k y / a y ) [ 177 ]2.75 10 6 96 PropertiesxGnP a T 10 6 ( m 2 s ) [ 182 ] 15-asmade 2.75, 15-annealed 3.28, 15-annealed,coldpressed 1.59, 1-as made 1.1, 1-anneled 1.39, 1-annealed,coldpressed 0.71 a jj 10 6 ( m 2 s ) [ 182 ] 15-asmade 188, 15-annealed 204, 15-annealed,coldpressed 215, 1-as made 17.2, 1-anneled 20.2, 1-annealed,coldpressed 24.5 k T ( J mKs ) [ 182 ] 15-asmade 1.43, 15-annealed 1.331, 15-annealed,coldpressed 1.28, 1-asmade 0.56, 1-anneled 0.7, 1-annealed,coldpressed 0.65 k jj ( J mKs ) [ 182 ] 15-asmade 98, 15-annealed 107, 15-annealed,coldpressed 178, 1-as made 8.66, 1-anneled 10.5, 1-annealed,coldpressed 22.6 t flux q 0 HT 0 ( k T / a T ) 15-asmade 52 10 4 , 15-annealed 40.5 10 4 , 15-annealed,coldpressed 80.5 10 4 , 1-asmade 50.9 10 4 , 1-anneled 50.35 10 4 , 1-annealed,cold pressed 91.5 10 4 ( k jj / a jj ) [ 182 ] 15-asmade 52.1 10 4 , 15-annealed 52.45 10 4 , 15-annealed,cold pressed 82.79 10 4 , 1-asmade 50.34 10 4 , 1-anneled 51.98 10 4 , 1- annealed,coldpressed 92.24 10 4 r c ( kg m 3 ) r 0 ( kg m 3 ) [ 182 ] 15-asmade 730, 15-annealed 730, 15-annealed,coldpressed 115, 1-as made 710, 1-anneled 710, 1-annealed,coldpressed 1300 c p ( J kgK ) [ 182 ]710.0 Table5.4:MaterialvaluesofxGnP. 97 PropertiesxGnPcomposite E ( GPa ) Figure7of[ 184 ] xGnP-1-PPnanocomposite 1.3-1.9, xGnP-15-PP nanocomposite 1.3-1.5,Figure3[ 185 ] xGnP/LLDPEnanocomposites 0.2-1.0 [ 186 ] monolayergraphenemembrane 1000 s c ( MPa ) Figure6of[ 184 ]strength xGnP-1-PPnanocomposite 40-51, xGnP- 15-PPnanocomposite 40-45,Figure3[ 185 ] xGnP/LLDPEnanocomposites 2-18 [ 186 ] monolayergraphenemembrane 130000 n [ 186 ] graphiteinthebasalplane 0.165 P 9 ( s c / E ) [ 184 ] xGnP-1-PPnanocomposite 0.0307-0.0268, xGnP-15-PPnanocomposite 0.0307-0.03,[ 185 ] xGnP/LLDPEnanocomposites 0.01-0.018 [ 186 ] monolayergraphenemembrane 0.13 Table5.5:MaterialvaluesofxGnPcomposites. Fromthevaluesofthermalconductivity,gravityandheatfordiffer- entplasticsintable 5.6 ,theirthermaldiffusivitiesarecalculated,yieldingrelativelylow values,whichareoforder10 8 10 7 .Belowtheirglasstransitiontemperature T g , mostplasticshavetensilemodulusofabout2 GPa andtensilestrengthlessthan35 MPa at roomtemperature[ 187 ].Thermosetsandthermoplasticsbehaviorsaredifferentbecause oftheirdifferentbondstructures.Particularly,thermoplasticshavebothweakandstrong bondsforwhichtheweakonesbreakwhenthematerialsareheatedwhilethestrongones arestillremained.Thermosetshaveonlystrongbondwhichwillbreakandthematerial decomposewhensubjectedtohightemperature.Thermosetmaterialshavehigherratio oftensilestrength s c vstensilemodulus E (intherange0.01-0.03)whencomparedwith thermoplastics(10 4 3 10 4 ). 98 PropertiesThermoset&Thermoplastics a from[ 187 ] polyurethaneresin 4.927 10 8 1.468 10 7 , urethaneelas- tomer 2.94 10 8 1.60 10 7 , urethanerigidfoam 6.69 10 8 1.53 10 7 , allylresin 6.18 10 8 7.02 10 8 , PMMA,commercialgrade 1.153 10 7 , PTFE 8.52 10 8 , Urea-fomaldehydealphacellulose 1.13 10 7 1.738 10 7 , polypropylene 0.95 10 7 , polystyrene 1.11 10 7 , polymethylmethacrylate 1.18 10 7 , polyvinylchloride 1.25 10 7 , polyethyleneterephthalate 1.43 10 7 k ( J mKs ) [ 188 ] uncuredandcuredthermosetpolyesters 0.106-0.2092, polypropylene 0.24 [ 187 ] polyurethaneresin 0.17-0.21, urethaneelastomer 0.07-0.3, urethane rigidfoam 0.06-0.12, allylresin 0.199-0.21, PMMA,commercialgrade 0.2, Urea-fomaldehydealphacellulose 0.285-0.409 Table5of[ 187 ] selectedplastics from0.12 UP to0.42 MF r c ( kg m 3 ) r 0 ( kg m 3 ) [ 189 ] benzoxazineepoxycopolymer 1200 [ 187 ] polyurethaneresin 1100.0-1500.0, urethaneelastomer 1100.0-1250.0, urethanerigidfoam 560.0-640.0, allylresin 1300-1400, PMMA,commercial grade 1180-1190, polyesterresin 1100-1460, Urea-fomaldehydealphacellulose 1480-1500 T a ( K ) A ( 1 s ) [ 190 ] phenolic 4.48 10 9 , nylon 1.85 10 13 [ 171 ] Hydrogenatedpolystyrene 1.4 10 14 , Poly-n-methylstyrene 7.2 10 16 , Poly- a -methylstyrene 8.3 10 18 Table5.6:Materialpropertiesofthermosetsandthermoplastics. 99 Table 5.6 (cont'd) E a ( kJ mol ) [ 190 ] phenolic 170.0, nylon 220.0 [ 171 ] Hydrogenatedpolystyrene 217, Poly-n-methylstyrene 246, Poly- a - methylstyrene 242 ˘ 200 t chem 1 A e E a RT 0 ! [ 190 ] phenolic 8.9 10 19 , nylon 1.09 10 25 ,[ 171 ] Hydrogenatedpolystyrene 4.34 10 23 , Poly-n-methylstyrene 9.47 10 25 , Poly- a -methylstyrene 1.65 10 23 ˘ 10 20 E ( GPa ) [ 191 ] AcrylonitrileButadieneStyrene 1.7-2.7, PBT 5-9, PA6 5-9 [ 192 ] epoxy 2.0, compositeepoxygraphene 3.1at300K, epoxy 5.9, composite epoxygraphene 7.4at77K,[ 193 ]3.24 [ 189 ] benzoxazineepoxy 0 50% copolymer,modulus 4.5-3.5 [ 187 ] glassreinforceEP 175.0, alpha-cellulose , MF 9.0, PF 5.0-7.0, wood PF6.0-8.0, glasspolyester 11.0-14.0 [ 194 ] Epoxyclay 0 5% nanocomposite 2 3,[ 195 ] glassreinforced epoxyatvariousstrainrates 37.4 41.7,[ 196 ] anhydridecuredepoxy 3.2 Table8of[ 187 ]thermosets Phenolics 6.9-20.7, Unreinforcedpolyesters 2.83-3.45, Unreinforcedepoxy 2.7-3.4, Reinforcedpolyesters 5.5-11.7 Table8of[ 187 ]thermoplastics PEEK 1.1, Polycarbonate 2.3, PEI 3.0, PES 2.6, PSU 2.48, PPE 2.5,ABS1.8-2.5, Nylon62.6 , Nylon6/6 1.59-3.79 Table10of[ 187 ]selectedthermoplasticswithglass0 40%,in- cludingStyrene,SAN,ABS,PP,glass-coupledPP,PE,AC,Polyester, Nylon6,Nylon6/6,Nylon6/12,PC,PSU,PPS,varyingfrom0.13( pure Polypropylene,pureNylon6/6 )to1.24( SAN 40% glass ) ˘ 5.0forthermosets,1.0forthermoplastics 100 Table 5.6 (cont'd) s c ( MPa ) [ 191 ] AcrylonitrileButadieneStyrene 19.6-49.0,PBT70-110,PA690-160, [ 189 ] benzoxazineepoxy 0 50% copolymer,strength 125-165 [ 187 ] glassreinforceEP 350.0, alpha-celluloseMF 50.0-90.0,PF 50.0-55.0, woodPF 45.0-60.0, glasspolyester 35.0-65.0,[ 194 ] Epoxy-clay 0 5% -nanocomposite 50 70,[ 195 ] glassatvari- ousstrainrate 784.5-1198,[ 196 ] anhydridecuredepoxy,thestressatbrittle rupture 40.0 Table1of[ 187 ] LDPE 10-12, HDPE 26-33, LLDPE 15-32, PPori-PP 31-37, TPX 28, PSora-PS 50, s-PS 41, PMMA 70, PVC 55, PVF 66-131, PVDF 48, PCTFE 30-39, PTFE 17-21, PVOH 83-152, POM 70, PEO 13-22, nylon11 38, nylon12 45, nylon4/6 100, nylon6/6 80, nylon6/10 55, PC 62, PET 72, PBT 52, PEI 105, PAI 152, PI 72-118, PSUorPSF,PAS,PPS 70, PEK 110 Table8of[ 187 ]thermosets AminosUF 38-48, AminosMF 48-55, PUR 24, Unreinforcedpolyesters 40-55, Reinforcedpolyesters 124-152, Unreinforced epoxy 42.7-82.7, Unreinforcedpolyimide 38.6 Table8of[ 187 ]thermoplastics Acetal 60.7-68.9, Polyamides 80.7-94.5, PEEK 91.7, Polycarbonate 62-72.4, PEI 105, PES 84.1, PSU 70.3, PPE 53.8, PPS 65.5, PPS 40% glass 138, PET 62.1, PET 30% glass 150, ABS 32-45 Table10of[ 187 ]selectedthermoplasticswithglass0 40%,in- cludingStyrene,SAN,ABS,PP,glass-coupledPP,PE,AC,Polyester, Nylon6,Nylon6/6,Nylon6/12,PC,PSU,PPSatroomtemperature32 ( Polypropylene )to214( Nylon6/6 40% glass ) n [ 196 ] anhydridecuredepoxy 0.4 ( 50 o C ) ,0.41 0.42 ( TT g = 30 o C ) ,0.5 ( T > T g ) 101 Table 5.6 (cont'd) c p ( J kgK ) [ 188 ] thermosetpolyesters 1673-1882 [ 187 ] polyurethaneresin 1300-2300, urethaneelastomer 1700-1900, urethane rigidfoam 1400,ofpolymersingeneral1250-2510, polystyrene 1170, PMMA,commercialgrade 1470, Urea-fomaldehydealphacellulose 1680 P 9 ( s c / E ) [ 191 ] AcrylonitrileButadieneStyrene 0.0115-0.0181,PBT0.014-0.012,PA6 0.018-0.0177 from[ 194 ] Epoxy-clay 0 5% -nanocomposite 0.025(pureepoxy)-0.035 (epoxy-5%clay),from[ 195 ] glassatvariousstrainrate 0.021 0.029 Table10of[ 187 ]selectedthermoplasticswithglass0 40%,in- cludingStyrene,SAN,ABS,PP,glass-coupledPP,PE,AC,Polyester, Nylon6,Nylon6/6,Nylon6/12,PC,PSU,PPSvaryingfrom0.096 10 3 (SAN 40% glass to0.305 10 3 (Nylon6/12) ˘ 0.02forthermosets,0.0002forthermoplastics P 7 ( r c / r 0 ) 0.0(plasticsareconsiderednon-charringmaterials) P 8 H 2 E r 0 A 2 t hc H 2 ( a 1 ) ˘ 10 7 t flux q 0 HT 0 ( k / a ) from[ 187 ] polyurethaneresin 1430000-3450000, urethaneelastomer 1870000-2375000, urethanerigidfoam 784000-896000, PMMA,commer- cialgrade 1734600-1749300, Urea-fomaldehydealphacellulose 2486400- 2520000 ˘ 2 10 6 t chem 1 A e E a 2 RT 0 ! [ 171 ] Hydrogenatedpolystyrene 55729.1, Poly-n-methylstyrene 36278.0, Poly- a -methylstyrene 141.1 102 Table 5.6 (cont'd) t chem 1 A e E a RT 0 ! [ 171 ] Hydrogenatedpolystyrene 4.34 10 23 , Poly-n-methylstyrene 9.4758743 10 25 , Poly- a -methylstyrene 1.6534333 10 23 TableIVin[ 171 ]containskineticsparametersforvariousorganicmaterials(ferulic acid,perylenetetracarboxylicacidanhydride,protocatechuicacid,naphthalenetetracar- boxylicacid,melliticacid,tartaricacid,polystyrene,polyethylene,hydrogenated polystyrene,poly-n-methylstyrene,poly-oc-methylstyrene,poly(methylmethacrylate), poly(methylacrylate),cellulose). 103 Frommaterialpropertiesintables 5.2 , 5.3 , 5.4 , 5.5 , 5.6 .,therepresentativeparameterval- uesforeachmaterialaretakenandtabulatedintable 5.7 .Thelastcolumncontainsmate- rialpropertyvaluesusedinthenumericalsimulationofsections 3.3 and 3.4 . RubbersCellulosicsThermosetsThermoplasticsOriginaldata a ( m 2 / s ) 10 7 3 10 7 10 7 10 7 k(J/mKs)0.10.3(z),0.2( r q )0.3 r c 400.0(kg)20% r 0 0.00.0300.0 r 0 ( kg ) 1000.0700.01200.01200.01000.0 A10 10 10 8 e 31.25 E a ( kJ / mol ) 100.0150.0220.0220.077 E ( GPa ) 1.015.05.01.0 s c ( MPa ) 14.080.0100.00.024 E n 0.490.40.40.45 c p 2000.01500.0 Table5.7:Representativematerialparameters. 104 CHAPTER6 CASESTUDIES Inthischapter,Iwillinvestigatehowvariousparametersaffecttheproblembyvarying eachofthem.Therectangulardomainasinchapter 3 isconsidered.Asmentionedin section,theproblemsconsistsoftotalparameters,whichare L , H , a x , a y , T 0 , q 0 k y , l , r c , A , T a , r 0 , s c , E , n .Amongtheseparameters,thesevenfollowingonesarevaried a x , a y , q 0 k y , l , A , T a , s c .OnlySection 6.1 isdrivenbythesamethermalpyrolysisproblemas Section 3.4 becauseonlythetensilestrength s c isvaried.Whileinothersections,thether- malandpyrolysisproblemarealsochangedbyvaryingotherparameters.Theoriginal valueofeachparameterwhicharelistedinEq.( 3.35 )willbedenotedwiththesuperscript 0 tobedifferentiatedfromitsvaluesthatarevaried. 6.1Theeffectofthecrackingthreshold Itisapparentthatmaterialwithlowertensilestrength s c ismorepronetocracking, asrinthethermalshockparameters f T = k s c ( 1 n ) g t E (Eq.( 1.4 )).Thissection investigateshow s c scaleswithquantitiesofcrackmorphologysuchascrackspacing, crackinitiation,totalopeningsurfaceandloopquantity.Itisexpectedthatthereduction oftensilestrengthleadstoearlierinitiation,denserandmoretotalcrackingsurface.In Fig. 6.1 ,thecrackpatternsandthemaximumprinciplestress s 1 at3000 s areplotted forthesimulationsusingdifferent s c ,inwhichvaluesareindicatedintheecaption. Thecrackpatternsat t = 3000,5000,6000 s arealsoplottedonthebackgroundwhichis thedensityforbettervisibilityinFig. 6.2 , 6.3 , 6.4 .Itcanbeseenfromthesees that,forsmall s c / E ,crackpatternshaveloopregions,whilesampleoflarger s c / E forms welldeveloped,tree-likebranches. Effectoncrackinitiationtime. 105 Figure6.1:Maximumprinciplestress s 1 at t = 3000 s correspondingtovariousvalues oftensilestrength s c / E ,whichequal0.024,0.0333,0.0467,0.0600,0.0667,0.0867,0.1000, 0.1133,0.1267,0.1333,0.1600,0.1733fromlefttoright,toptobottom,respectively. Thehigherthetensilestrength s c ,thelongerittakesforthesampletoreachthatvalue andthustoinitiatethecrack.Fig. 6.5 leftshowshowthecrackinginitiationtime t i changeswithtensilestrength,inwhich s c isrescaledby s m thustherescaledtensile strength s c / s m variesbetween0and1.Therightlimitvalue1correspondstomaterial 106 Figure6.2:Crackingpatternforvariousvaluesoftensilestrength s c at3000 s .As s c getslarger,crackmorphologyshiftsfromloop-liketowardwelldevelopedtree-likeor branchingbehavior. thatwillnevercrack.Thus,inprinciple,thecurveshouldasymptotetheverticalline s c / s m = 1.However,thisbehaviorisnotseenfromFig. 6.5 left,probablyduetothe sizeoftheelements.For s c / s m thatislessthan0.8,the t i vs. s c / s m curvefollows powerlawoftime.TheredlineinFig. 6.5 leftisthebestleastsquaretothedata.Part 107 Figure6.3:Crackingpatternforvarioustensilestrengthat5000 s .As s c getslarger,crack morphologyshiftsfromloop-liketowardwelldevelopedtree-likeorbranchingbehavior. ofthedatainFig. 6.5 leftcorrespondingto s c / s m < 0.8isreplottedonalog-logscalein Fig. 6.5 rightforbetterobservationofthepowerbehavior. Theeffectontotalcracklength Beforeanycrackinitiatesinthesample,thesampleisexposedtotheincidentheat 108 Figure6.4:Crackingpatternforvarioustensilestrengthat6000 s .As s c getslarger,crack morphologyshiftsfromloop-liketowardwelldevelopedtree-likeorbranchingbehavior. throughtheheatlength l .Whencracksappear,theyopenupthesampleinterior.The totalsurfaceopennedupduetocrackingischaracterizedbythetotalcracklength l c .It isaveryimportantlinkintheviciouscycleofthethermaldegradationprocess.Fig. 6.6 plotstheevolutionofthesampletotalcracklength l c ofvarious s c indicatedinthelegend box. 109 Figure6.5:Left:Crackinitiationtimevstensilestrength.Left:Partoftherightethat correspondsto s c / s m < 0.8plottedinalog-logscale. Figure6.6:Evolutionoftotalcracklengthforvarioustensilestrength. Itcanbeseenfromtheplotthat,formostsamples,thetotalcracklength l c insamplewith lower s c isshorter,providedthatthecomparisonacrossdifferentsamplesismadeatthe sametime.EachcurveinFig. 6.6 qualitativelyconsistsoftwoparts:thenonlinearpart neartheignitiontimeFig. 6.7 andthelinearpartwithslopes b decreasingconsistently with s c . Fig. 6.8 (c)plots b against s c inalog-logscale.ThenonlinearpartsofthecurvesinFig. 6.7 formostvaluesof s c areintotheequation 110 Figure6.7:NonlinearpartoftheFig. 6.6 . log ( l c / l )= alog ( t t i )+ b (6.1) asshownbyFig. 6.8 (b),whichmeans l c ispowerintime,forwhichvaluesof a and b vs s c / s m areshowninFig. 6.8 (d).Samplewith s c / s m nearunityisnotintothe powerEq.( 6.1 ). Theeffectoncrackspacing AsseenfromFig. 6.9 ,ingeneral,crackspacingislargerintoughermaterials.Allcurves inFig. 6.9 havesimilartrend:aquickdroppingfollowedbyaconstantvalueperiodand thenarisingperiod.Andjumpsinspacingduetocrackbranchingorintersectingisalso largerbecausematerialswithlargertensilestrength s c havefewercracks,thusincreasing ordecreasingthenumberofcracksjustbyonemaycausessubstantialchangeintheav- eragespacing. Moreover,tensilestrengthalsoaffectscracksegmentlengthandjunctionangles.As s c increases,cracksegmentsgetlonger,thenumberofrightanglejunctionsdeclinedueto 111 Figure6.8:(a)Fig. 6.7 withtimeaxisofeachcurveshiftedbyitsinitiationtime t i .(b) a , b areparametersofthecurve log ( l c / l )= alog ( t t i )+ b inEq.( 6.1 ).(c)slope b ofthelinearpartsvs.tensilestrength s c / s m onaloglogscale.(d)valuesof a and b vs. s c / s m . Figure6.9:Averagecrackspacing g vsdepth r c / l forvarious s c / E asindicatedinthe legend. morebranchingthanintersection.Theprobabilitydistributionsofsegmentlengthand junctionanglesforvarious s c valuesareplottedinFigs. 6.10 and 6.11 . Theevolutionofthetotalnumberofloopsanditsdependenceonthethetensilestrength 112 Figure6.10:Probabilitydistributionofsegmentlengthwhenthecrackimageisnot thresholded(left)andthresholded(right).Differentcolorscorrespondtodifferentval- uesof s c / s m asindicatedinthelegend. Figure6.11:Probabilitydistributionofjunctionanglewhenthecrackimageisnotthresh- olded(left)andthresholded(right)forvariousvaluesofthetensilestrengh s c / s m as indicatedinthelegend. canbeseenfromFig. 6.12 . 6.2Theeffectofthermaldiffusivities AsdiscussedpreviouslyinSection 3.4 ,crackstendtopropagateinthedirectionsthat areperpendiculartothecharfront.Thustheprofcracknetworkisdecidedbythe shapeofthecharfront,whichisdirectlyrelatedtotheanisotropyofthethermaldiffusiv- ities,i.e.,theratioof a y and a x or P 3 group.Also,materialsofhigherthermaldiffusivity 113 Figure6.12:Numberofloopschangeswithtimefordifferentvaluesof s c / s m (left)and itsdependenceon s c / s m at t = 1000,2000,3000,4000,5000,6000 s (right). allowheattodiffusemorequickly,thushavelowerthermalgradientaswellasdensity gradientandstresses.Excellentmaterialswithhighthermalanisotropyaregraphiteand theirderivativessuchasxGnPgraphenenanoplatelets.Theindepththermaldiffusivity a y facilitatesheatpenetratingintothesample,thuslarger a y increasescrackingdepth intothesampleinterior.Ontheotherhand,thehorizontalthermaldiffusivity a x facili- tatescrackpropagationhorizontally,reducingcrackingdepth. Thedistributionofthedimensionlesstemperature q alongtheverticalmiddlelineofthe sampleat t = 100,500,1000,1500,2000,2500,3000 s canbeseenfromFig. 3.3 forthree casesof a x .As a x getslarger(Fig. 3.3 (c)),thetemperaturemorequicklyreachesthestate ofequilibria.Futhermore,thesteadystatevalueoftemperaturecorrespondingtohigher a x issmallerandviceversa. Theeffectsof a x onthedensityisillustratedinFig. 6.14 . Theevolutionoflocations y ofthecharfront r = 0.3(solid),thepyrolysisfront r = 0.99 (dashed)andtheirdifference d (dotted)forthreecasesof a x (equal0.1,1,1oftheoriginal valuerespectively)isindicatedinFig. 6.15 .Itcanbeseenthatsamplesoflower a x get 114 Figure6.13:Thedistributionofthedimensionlesstemperature q alongtheverticalmiddle lineofthesamplewhere x = 0.5atdifferenttimesasindicatedinthelegendforthree casesof a x .Left: a x = 0.1 a 0 x .Middle: a x = a 0 x .Right: a x = 10 a 0 x . Figure6.14:Theeffectsof a x ondensityesaretakenat t = 3000 s .Lefte: a x = 0.1 a 0 x , P 3 = 10 P 0 3 .Righte: a x = 10 a 0 x , P 3 = 0.1 P 0 3 . indepthcharredmorequicklywhichimpliesthecharfrontandthepyrolysisfrontreach thelowersurfaceatafasterrate.Thepyrolysislength d for a x = 0.1 a 0 x hascomparable valuewiththeoriginalcase,butitquicklydropstonearzeroasthecharfrontapproaches thelowersurface. Thecrackingpatternandmaximumprincipalstress s 1 forthreecasesof a x areshownby Fig. 6.20 .Thesampleoflower a x (Fig. 6.16 left)initiatesthecrackat56 s whilethe sampleofhigher a y (Fig. 6.16 right)takes385 s tonucleatetheone. While a x onlyaffects P 3 , a y alsoaffectstheheatconductioncharacteristictime t hc = H 2 / a y ,andthusthegroup P 6 = t hc / t chem = H 2 Aexp ( T a / T 0 ) a y .Thedistributionsof thedimensionlesstemperature q alongtheverticalmiddlelineofthesampleat t = 100, 500,1000,1500,2000,2500,3000 s areplottedinFig. 6.17 forthreecasesof a y .Similarto a x ,higher a y (Fig. 6.17 (c))alsocausesthetemperaturetomorequicklyreachthesteady state.Futhermore,thesteadystatetemperaturecorrespondingtohigher a y issmallerand 115 Figure6.15:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolidline. Thelocationofthepyrolysisfrontwhere r = 0.99isindicatedbythedashedline.The pyrolysislength d isindicatedbythedottedline. Figure6.16:Varying P 3 = a y / a x viachanging a x .Lefte: a x = 0.1 a 0 x , P 3 = 10 P 0 3 , cracksgrowindepth.Righte: a x = 10 a 0 x , P 3 = 0.1 P 0 3 ,cracksspreadhorizontally. moreuniformlydistributedalong y ,thustheonsetofpyrolysisandcrackinitiationare delayed.However,oncecracksinitiate,theywillquicklyevolveandreachthelowersur- facebecausetherateofindepthheattransferandpyrolysisarehigherforlarger a y . Figure6.17:Thedistributionofthedimensionlesstemperature q alongtheverticalmiddle lineofthesamplewhere x = 0.5atdifferenttimesasindicatedinthelegendforthree casesof a y .Left: a y = 0.1 a 0 y .Middle: a y = a 0 y .Right: a y = 10 a 0 y . 116 Theeffectsof a y onthedensityisillustratedinFig. 6.18 .Itcanbeseenfromthese esthatthesampleofhigher a y samplehasthickerpyrolysislength d andthuslarger crackspacing. Figure6.18:Varyingboth P 3 = a y / a x and P 6 = H 2 Aexp ( T a / T 0 ) / a y byvarying a y , Densityat t = 2350 s .Leftre: a y = 0.1 a 0 y leadsto P 3 = 0.1 P 0 3 , P 6 = 10 P 0 6 . Righte: a y = 10 a 0 y leadsto P 3 = 10 P 0 3 , P 6 = 0.1 P 0 6 . Theevolutionoflocations y ofthecharfront r = 0.3(solid),thepyrolysisfront r = 0.99 (dashed)andtheirdifference d (dotted)forthreecasesof a y (equal0.1,1,1oftheoriginal valuerespectively)isindicatedinFig. 6.19 . Figure6.19:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolidline. Thelocationofthepyrolysisfrontwhere r = 0.99isindicatedbythedashedline.The pyrolysislength d isindicatedbythedottedline. Thecrackingpatternandmaximumprincipalstress s 1 forthreecasesof a y areshownby Fig. 6.20 .Thesampleoflower a y (Fig. 6.20 left)initiatesthecrackat9 s whilethe 117 sampleofhigher a y (Fig. 6.20 right)takes1090 s tonucleatetheone. Figure6.20:Varyingboth P 3 = a y / a x and P 6 = H 2 Aexp ( T a / T 0 ) / a y byvarying a y , t = 2350 s .Lefte: a y = 0.1 a 0 y leadsto P 3 = 0.1 P 0 3 , P 6 = 10 P 0 6 ,cracksgrow horizontally.Righte: a y = 10 a 0 y leadsto P 3 = 10 P 0 3 , P 6 = 0.1 P 0 6 ,cracksgrowin depthandtherearefewercracks. Thus,fromvaryingthethermaldiffusivities a x and a y ,itcanbeconcludedthathigher thermaldiffusivitycausesthetemperaturetoreachtheequilibriastatemorequicklyat alowervalueandalsodelaysthepyrolysisaswellascrackingprocess.However,once charringorcrackingstart,theprocessesevolveatafasterrate.Moreover,theratio a y / a x orgroup P 3 decidestheisotropyofthecharregion:higher P 3 causeselongationofthe charregionintheverticaldirectionandmoreloopsformedinthecrackpatterns. 6.3Theeffectofheatstrength Inthissection,theproblemsofvariousheatstrength q 0 arestudied.Except q 0 , everyotherparametersremainthesameasindicatedbyEq.( 3.35 ).Asdiscussedinthe previoussection,Section 6 , q 0 rescaleswiththedepthofheatpenetration l f .Alsohereit isnotedthatchangingtheheat q 0 willaffectthefollowingdimensionlessPigroup, namely P 4 = t hc / t flux = q 0 H / k y T 0 bychangingtheheatcharacteristictime t flux = HT 0 k y a y q 0 .Inaddition,itcanbederivedanalyticallythat,thedimensionlesstemperature q scaleslinearlywith q 0 . Thedistributionsofthedimensionlesstemperature q alongtheverticalmiddlelineof thesampleat t = 100,500,1000,1500,2000,2500,3000 s areplottedinFig. 6.21 forthree casesof q 0 asindicatedintheecaption.Thegeneraltrendasexpectedishigherheat 118 correspondtohighertemperatureandviceversa.Futhermore, q islinearlypropor- tionalto q 0 . Figure6.21:Thedistributionofthedimensionlesstemperature q alongtheverticalmiddle lineofthesamplewhere x = 0.5atdifferenttimesasindicatedinthelegendforthree casesof q 0 .Lefte: q 0 = 0.1 q 0 0 .Middlee: q 0 = q 0 0 .Righte: q 0 = 10 q 0 0 . q scaleslinearlywith q 0 . Theeffectof q 0 onthedensitycanbeseenfromFig. 6.22 . Figure6.22:Density r at3000 s .Varyingheat q 0 affects P 4 = q 0 H / ( k y T 0 ) .Left e: q 0 = 0.5 q 0 0 , P 4 = 0.5 P 0 4 .Righte: q 0 = 2 q 0 0 , P 4 = 2 P 0 4 . Theevolutionoflocations y ofthecharfront r = 0.3(solid),thepyrolysisfront r = 0.99 (dashed)andtheirdifference d (dotted)forthreecasesof q 0 (equal0.5,1,2.0oftheoriginal valuerespectively)isindicatedinFig. 6.23 .Itcanbeseenfromthisethat d isnot muchaffectedbythechangeof q 0 when t < 1.5 t hc .However,thesamplepyrolyzesmore quicklywhen q 0 israised. Themagnitudeof q 0 affectstherateofpyrolysisandthustherateofcrackpropagation butdoesnotaffectthecrackspacing,asseenfromFig. 6.24 ,whichshowsthedistribu- tionsofthemaximumprincipalstresses s 1 correspondingtothreevaluesof q 0 indicated 119 Figure6.23:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolidline. Thelocationofthepyrolysisfrontwhere r = 0.99isindicatedbythedashedline.The pyrolysislength d isindicatedbythedottedline. inthecaption. Figure6.24:Maximumprinciplestressat3000 s .Varyingheat q 0 affects P 4 = q 0 H / ( k y T 0 ) .Lefte: q 0 = 0.5 q 0 0 , P 4 = 0.5 P 0 4 .Righte: q 0 = 2 q 0 0 , P 4 = 2 P 0 4 . ItisinterestingtonoteaboutthesimilaritybetweenthecrackpatternsinFig. 6.24 (c) ( q 0 = 2 q 0 0 )at3000 s andFig. 6.26 ( q 0 = q 0 0 )at6000 s (fromSection 3.4 );betweenthecrack patternsinFig. 6.24 upper( q 0 = 0.5 q 0 0 )at3000 s andFig. 6.27 ( q 0 = q 0 0 )at1500 s (from Section 3.4 ),intermofcrackdepthandcrackspacing. 120 Figure6.25:Theeffectof q 0 oncrackspacing. Figure6.26:Maximumprinciplestress(fromSection 3.4 )when q 0 = q 0 0 at6000 s .Crack morphologyinthiseissimilartothatofFig. 6.24 lower Figure6.27:Maximumprinciplestress(fromSection 3.4 )when q 0 = q 0 0 at1500 s .Crack morphologyissimilartothatofFig. 6.24 upper. 6.4Theeffectofheatedregionsize 6.4.1Generalobservations Inthissection,theonlyparameterthatvariesisthedimension l oftheregionoverwhich heatisapplied.Thescalevariesthroughonlythisparameter,indicatingthat 121 thismodelcanpotentiallybeadaptedtoproblemsthatspantherangebetweenverysmall (micrandverylargeThisparameter l theproblemun- dertheeffectofthedimensionlessPigroup P 2 whichisitsratiowiththedomainlength P 2 = l / L .Thechangeofcrackmorphologycorrespondingtochanging l iscompara- bletothenonlinearfeedback:themoresurfacesexposedtohotgases,themorequickly materialsbecomecharandmorecracksdevelop. Theof l onthedensitycanbeseenfromFig. 6.28 ,inwhichthelefte correspondsto l = 0.01 L andtherightecorrespondsto l = L .Themiddlee correspondstothestandardcasewhen l = 0.1 L .Intheuppere,theactivationen- ergyisalsoraised100timeshigherthanitsstandardvaluetoacceleratethepyrolysisrate andinducecrack.However,onlyasmallcrackisobservedat t = 6000 s ,seeFig. 6.29 right.Larger l causeslargerareaofthesampletobequicklypyrolyzedandcharred. Figure6.28:Effectsofvaryingthesizeofheatedregion l andthus P 2 = l / L ondensity r .Lefte: P 2 = 0.01, A = 100 A 0 , t = 6000 s .Middlee: P 2 = 1.0, t = 3000 s .Righte: P 2 = 0.1, t = 3000 s Figure6.29:Effectsofvaryingthesizeofheatedregion l oncrackmorphology.Lefte: P 2 = 0.01, A = 100 A 0 , t = 6000 s .Middlee: P 2 = 1.0, t = 3000 s .Righte: P 2 .1, t = 3000 s . Therateofpyrolysisandthepyrolysislength d fortwocasesof l ,asindicatedinthe elegend,isshownbyFig. 6.30 . 122 Figure6.30:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolidline. Thelocationofthepyrolysisfrontwhere r = 0.99isindicatedbythedashedline.The pyrolysislength d isindicatedbythedottedline. 6.4.2Theeffectofvarying s c when l = L Inthissubsection,thevariouscasessharethesametemperatureanddensitydistributions astheonlyparameterthatisvariedisthetensilestrength s c ,similartotheanalysisinSec- tion 6.1 ,exceptherethewholeuppersurfaceisheated,i.e, l = L .Thedensityat certaintimesindicatedintheecaptionareplottedinFig. 6.31 . When l = L ,ifthesampledoesnotdevelopcrack,themaximumpossiblevaluethatthe maximumprinciplestress s 1 canattainsis s m 2 = 0.2 E ,(comparedwith0.18666 E when l = 0.1 L inSection 3.4 ).Fig. 6.32 showsthecrackpatternsanddistributionofthemax- imumprinciplestress s 1 at t = 3000 s forvariousvaluesof s c asindicatedinthee caption. Inthiscaseofuniformlyheatingontheuppersurface, l = L ,thecrackspacingatcertain depthismeasuredbyplacingahorizontallineintothecrackpatternandcountingthe 123 Figure6.31:Dimensionlessdensity r = r / r 0 at t = 50,100,500,1000,1200,1350,3000, 6000 s when l = L .OtherparametervaluesaregivenbyEq.( 3.35 ).Ataround t = 30 s , theuppersurfacestartstopyrolyzeandataround1350 s ,sodoesthelowersurface. intersectionpixels.Thecrackspacinginthiscasehasunitoflength,insteadofradianlike inSection 3.4 .Fig. 6.35 illustratestheprocedureofmeasuringcrackspacingatthedepth of0.5 H inthiscase. AsseenfromFig. 6.35 ,theaveragecrackspacing s increasesquiteconsistentlywithdepth r c ,exceptfortwolowestvaluesofthetensilestrength s c ,inwhich s showson when r c getcloserto H (sampleheight.).Thesuddenjumpsoneachcurveindicatebifur- cation(jumpingup)orjoining(jumpingdown)ofcracksegments.Moreover,samplesof higher s c havelargercrackspacingandviceversa. 124 Figure6.32:Thecrackpatternsanddistributionofthemaximumprinciplestress s 1 at t = 3000 s for s c / E =0.0240,0.0333,0.0467,0.0600,0.0667,0.0867,0.1000,0.1733. Figure6.33:Crackspacingismeasuredbyplacingalineintothecrackpatternandcount- ingintersectionpoints.Inthise,thelineisplacedatdepth0.5 H . Thetimedependenceoftotalcracklength l c forvarious s c isshowninFig. 6.35 .Itcan beseenfromtheethat,eachcurvequalitativelyconsistsoftwoparts:thenonlinear partwhen t < 1.4 t hc andthelinearpartwhen t > 1.4 t hc .Forthecase l = L ,around 125 Figure6.34:Theaveragecrackspacing s asafunctionofdepth r c forvariousvaluesof s c asindicatedinthelegend. t = 1.4 t hc isthetimewhenthelowersurfacestartstobecomecharasseenfromFig. 6.31 . Atthistime,mostofthesamplehasbeencharred.When t > 1.4 t hc ,cracksslowlyevolve, whichisindicatedbythesmallslopesofthelinearparts.Besides,theseslopesdecrease withvaluesof s c :samplewithhigh s c almostdevelopnofurthercracksafterthisperiod. Figure6.35:Theevolutionoftotalcracklength l c forvariousvaluesof s c asindicatedin thelegendwhen l = L .Thetotalcracklengthisrescaledbythelength l oftheheated region,whichisthesamplewidthinthiscase. 126 Figure6.36:Initiationtime t i ( s ) changeswithtensilestrength s c when l = L ,inwhich s c isrescaledbythemaximumstressvalueofsamplewithoutcracks s m 2 = 0.2 E . 6.5Theeffectofactivationenergy Whiletheparametersthatarevariedintheprevioussections,namely,thethermaldiffu- sivities a x , a y ,theheat q 0 ,thesizeoftheheatedregion l affecttheproblembyviathe temperature,inthesenexttwosections,theactivationenergy T a andthepre-exponent factor A oftheArrheniusequation( 2.2 )arevaried.Thusthetemperaturedistribution remainsthesameasthestandardcases.Onlythedensityandthusthecrackingbe- haviorareSince T a appearsinsidetheexponentialterm,itisonlyvariedby 10%ofitsoriginalvalue. Theevolutionoflocations y ofthecharfront r = 0.3(solid),thepyrolysisfront r = 0.99(dashed)andtheirdifference d (dotted)forthreecasesof T a (equal0.9,1,1.1 oftheoriginalvaluerespectively)isindicatedinFig. 6.38 . Theactivationtemperature T a affectsthechemicalcharacteristictime t chem = A 1 exp ( T a / T 0 ) andtwoPigroups P 5 and P 6 whichareinSection 6 as P 5 = T a / T 0 and P 6 = t hc / t chem = H 2 Ae T a / T 0 / a y throughtheeffectof P 5 .Substanceswithhigheractivation 127 Figure6.37:Varyingactivationtemperature T a affects P 5 = T a / T 0 and P 6 = H 2 Ae T a / T 0 / a y .Left: r plots.Right: s 1 plots.Uppere: T a = 0.9 T 0 a leadsto P 5 = 0.9 P 0 5 , P 6 = 23.336 P 0 6 .Lowere: T a = 1.1 T 0 a leadsto P 5 = 1.1 P 0 5 , P 6 = 0.04285 P 0 6 . Figure6.38:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolidline. Thelocationofthepyrolysisfrontwhere r = 0.99isindicatedbythedashedline.The pyrolysislength d isindicatedbythedottedline. temperaturehavelargerchemicalcharacteristictimeandslowerpyrolysisratebecause higheramountofenergyisrequiredtostartthereaction.Theeffectof T a oncrackspac- 128 ingcanbeseenfromFig. 6.39 . Figure6.39:Theeffectof T a oncrackspacing. 6.6Theeffectofpre-exponentfactor Thepre-exponentfactor A hasunitoftime,itinverselyscaleswiththecharacteristic timeforpyrolysisreaction t chem = A 1 exp ( T a / T 0 ) andaffectsgroup P 6 = t hc / t chem = H 2 Ae T a / T 0 / a y .Effectsof A onthecrackpatterncouldbeseenfromFigs. 6.40 upper and 6.40 lowerinwhich A isincreasedbyafactorofteninFig. 6.40 upperanddecreased bythesamefactorinFig. 6.40 lower. Theevolutionoflocations y ofthecharfront r = 0.3(solid),thepyrolysisfront r = 0.99(dashed)andtheirdifference d (dotted)forthreecasesof A (equal0.1,1,10ofthe originalvaluerespectively)isindicatedinFig. 6.41 . 6.7Morphologicaldiagram Asmentionedpreviously,fracturepatternsbelongstoabroadertopologicalclass,the networkpattern.Theycanbeintoseveralgroupsbasedontheirmorpholo- gies.Treelikepatternsarecharacterizedbybranchingandbifurcation.Theyaretypically 129 Figure6.40:Varying P 6 = H 2 Ae T a / T 0 / a y byvarying A , t = 3000 s .Left: r plots. Right: s 1 plots.Upperures: A = 10 A 0 . P 6 = 10 P 0 6 . t = 3000 s .Lowerse: A = 0.1 A 0 . P 6 = 0.1 P 0 6 . t = 3000 s . Figure6.41:Thelocationofthecharfrontwhere r = 0.3isindicatedbythesolidline. Thelocationofthepyrolysisfrontwhere r = 0.99isindicatedbythedashedline.The pyrolysislength d isindicatedbythedottedline.Whilehighervalueof A acceleratesthe pyrolysis, d doesnotchangedrasticallywith A . seeninsystemsinwhichdirectionaltransportationanddeliverydominate,suchasblood capillaries,treebranches,rivernetwork,leafveins.Themeaningoftransportationcould 130 Figure6.42:Theeffectof A oncrackspacing.When A = 0.1 A 0 ,cracksevolveatamuch slowerpaceandonlymaketheirwaytohalfofthesampledepth.Thesuddenjumpof crackspacinginthiscaseat r c / l = 1.5iscausedbyperiodicdoublingseenatearlystage ofcrackevolution.Thecase A = 10 A 0 producessimilarmorphologytotheoriginalcase buttheevolutionhappensatafasterrate. beextendedtoheattransferordirectionalcharringinourproblem.Ontheotherhand, thepolygonalorinterconnected,looplikepatternsaremoredirectconsequencesofthe minimizationprinciple,whichistheminimizationoftotalenergyinfractureandofother quantitiesinothercontexts.Thedrivingofthesepatternsareisotropic,actinguni- formlyonasurfaceataninstantoftime.Examplesaremudcrack,skincrack,ceramic glaze,craze. AsdiscussedinChapter 4 ,thecrackmorphologycancharacterizedintermsofquan- titiessuchas:fractaldimension,segmentrank,crackspacing,cracklength,numberof loops,loopdirection,junctionangles.Thesequantitiesareaffectedbythevaluesofpa- rametervalues.Inparticular,threeparameters q 0 , A and T a ,affecttherateofheattransfer orpyrolysisandthuscrackelongation,butdonothavemuchonthecrackmor- phology.Theycanbeviewedastheparametersthatscalethecharacteristictimesofthe problem.Ontheotherhand,crackspacingisbytwogroups P 9 = s c / E and Pi 6 = H 2 Aexp ( T a / T 0 ) a y .Whenthethermaldiffusivityinonedirectionincreases,tem- peratureanddensitygradientsinthatdirectiongetsmallerwhilethepyrolysislength 131 d increases.Thisresultsinoverallsmallerstressesinthesampleandlargercrackspac- ing.Iffractureresistanceisunderstoodaslessanddelayeddamage,apyrolysiscracking parametercanbetakenas f P = a y exp ( T a / T 0 ) s c AE ,(6.2) whichistheratioof P 9 and P 6 .Similartothethermalshockparameter f T = k s c ( 1 n ) g t E (Eq.( 1.4 ))thatcharacterizesdamageinthermoelasticity,highervalueof f P corresponds tomaterialsthataremorefractureresistantinthismodel. Group P 9 notonlycharacterizescrackspacing,italsocharacterizethemorphologi- calshiftfromloop-liketotree-likebehaviorwhenitsvaluegetslarger.Thisbehavioris alsobythethermalanisotropyor P 3 group:when P 3 = a y / a x islarge,cracks clustertogethernearthecentralregionofthesample,formingloopnetwork.When P 3 is small,cracksspreadingout,formingtreelikestructure.ThechartbelowFig. 6.43 repre- sentsthecompetitiveofthesetwogroups P 3 and P 9 onthecrackmorphology intermoftreeandloopbehavior. 132 Figure6.43:Thecompetitiveof P 3 = a y / a x and P 9 = s c / E onthemorphol- ogyofthecrackpatterns.Thehorizontalaxisisthetensilestrength s c rescaledbyits maximumvalue s m .Theverticalaxisisonthelogarithmscale, log 10 ( a x / a y ) whichis log 10 ( P 3 ) .Closetotheoriginwhere a y / a x islargeand s c / E issmall,thecrackstend toformloops.Awayfromtheorigin,theytendtodevelopbranches,formingtree-like pattern. 133 CHAPTER7 CRACKSONACIRCULARDOMAIN 7.1Aradialheatingproblem Inthepreviouschapters,theproblemisconsideredonarectangulardomain.Ona circulardomainwiththeassumptionofaxialsymmetry,partialanalyticalsolutionscanbe derived.Itisflpartialflbecauseofthefactthat,whilethereexistsananalyticaltemperature andananalyticalexpressionforstressesintermsofdensity,theArrheniusequationwhich relatesdensitytotemperature,isnotintegrable.Inparticular,considerthefollowing heatingproblemofacylindricalrod: ProblemA Acylindricalrodofradius R andlength L ,initiallyatuniformtemperature T 0 , isheatedbyanuniformheatperunitlength q 0 fromitscenter r = 0.Thetemperature T isgovernedbytheheatconductionequation,Eq.( 2.1 )which,incylindricalcoordinate andunderaxis-symmetriccondition,becomes ¶ T ¶ t = a r r ¶ ¶ r r ¶ T ¶ r ,(7.1) where a r isthethermaldiffusivityintheradialdirection.Tosolvethe problemA ,let consideramoregeneralproblem ProblemB ,inwhichitsonlydifferencewith problem A isthatthecylindricalrodishollow.Let r i beitsinnerradiusand R itsouterradiusR. TheFourier'slawthatrelatesheattotemperaturegradientprovidestheboundary conditionatfor ProblemB at r = r 0 as 2 p r 0 k r ¶ T ¶ r j r = r i = q 0 ,(7.2) where k r isthethermalconductivityintheradialdirectionand q 0 istheheatreleaseper unitlengthL.When r i = 0, ProblemB isidenticalto ProblemA . 134 Theanalyticalsolutionofthe ProblemB isobtainedusingthemethodofsimilarvariable. Fromthat,theanalyticalsolutionofthe ProblemA isobtainedbytakinglimit r i ! 0, whichis T ( h ) T 0 q 0 / ( 2 p k r ) = Z ¥ h e s 2 s ds ,(7.3) where h = r / ( 2 p a r t ) isthesimilarityvariable.Thedensityofthecylindricalrodfollows fromtheArrheniusrelation,Eq.( 2.2 ),whichis ¶r ¶ t = A ( r r 0 ) e T a T .(7.4) inwhich r 0 istheinitialroddensity, T a istheactivationtemperatureand A isthepre- exponentfactor.Theseparametersarethesameasthosefrompreviouschapters.The stressbalanceequationinradialcoordinateunderaxissymmetricconditionistakenfrom Eq.( 2.8 )as d s rr dr + s rr s qq r = 0,(7.5) andthetotalstrainfollowEquations( 2.6 ),( 2.4 )fortheaxissymmetricproblemanda generalscalarshrinkagestrain # 0 : # rr = s rr ns qq E + # 0 , # qq = s qq ns rr E + # 0 .(7.6) Thesolutionofthestress-strainproblem( 7.5 )and( 7.6 )withtraction-freeboundarycon- dition s rr j r = R = 0,withoutcrackingisgivenas 135 s rr = E R 2 Z R 0 # 0 ( z ) z d z E r 2 Z r 0 # 0 ( z ) z d z , s qq = E # 0 + E R 2 Z R 0 # 0 ( z ) z d z + E r 2 Z r 0 # 0 ( z ) z d z .(7.7) ThedetailedderivationofsolutiongivenEq.( 7.7 )whichcorrespondstotractionfree conditionattheouterradiusandtheothersolutioncorrespondingtopinnedconditionis giveninSection 7.2 .Inourmodel,theshrinkagestrainistakenasproportionaltomass depletion # 0 = g ( r r 0 ) asalreadygiveninEq.( 2.5 ).Thusthesolution( 7.7 )becomes s rr = g E R 2 Z R 0 ( r r 0 ) z d z g E r 2 Z r 0 ( r r 0 ) z d z ,(7.8) forradialstressand s qq = g E ( r r 0 )+ g E R 2 Z R 0 ( r r 0 ) z d z + g E r 2 Z r 0 ( r r 0 ) z d z .(7.9) fortangentialstress.Since r < r 0 ,itcanbeproventhat s rr > 0, d s rr / dr < 0and s qq ( r = R ) < 0if d r / dr > 0.Viceversa, s rr < 0, d s rr / dr > 0and s qq ( r = R ) > 0if d r / dr < 0.Physically,thisimpliesthatifthesamplelosesitsmassfromitscenter,the radialstresswillbetensileanddecreaseswithradius.Inparticular,usingintegrationby partsthefollowingintegralbyletting z d z = d 1 2 z 2 Z r 0 ( r r 0 ) z d z =( r ( r ) r 0 ) r 2 2 Z r 0 z 2 2 d r d z d z ,(7.10) whenevaluatedat r = R ,giving Z R 0 ( r r 0 ) z d z =( r ( R ) r 0 ) R 2 2 Z R 0 z 2 2 d r d z d z .(7.11) SubstituingtwoaboverelationsintoEq.( 7.9 ),estimatingat r = R ,yielding s qq ( r = R )= 2 g E R 2 Z R 0 d r d z z 2 2 d z ,(7.12) 136 whichisnegativeas d r dr > 0.Moreover,takingthelimitas r ! r e ,where r e isverysmall, ofthefollowingquantity: Lim r ! r e 1 r 2 Z r 0 ( r r 0 ) z d z = Lim r ! r e ( r r 0 ) r 2 r (7.13) =( r ( r e ) r 0 ) , (7.14) thensubstitutingtheaboveexpressionandusing( 7.11 )intoEq.( 7.9 )thenevaluatedat r e = 0givesthevalueofthetangentialstressatthecenteras s qq ( r = 0 )=( r j ( r = R ) r j ( r = 0 )) g E R 2 Z R 0 d r dr z 2 2 d z ,(7.15) whichcanbebothpositiveornegativebecauseintheRHSof( 7.15 ),thetermis positiveandthesecondtermisnegativefor d r dr > 0.FromEq.( 7.8 ),takingthederivative ofitssecondterm d dr g E r 2 Z r 0 ( r r 0 ) z d z = 2 g E r 3 Z r 0 ( r r 0 ) z d z + g E r 2 ( r r 0 ) r ,(7.16) thus,thederivativeofradialstressis d s rr dr = g E r 3 Z r 0 d r d z z 2 2 d z ,(7.17) whichisalwaysnegativewhen d r dr > 0.Notethattheaboveexpressioncanalsobe obtainedfromEqs.( 7.5 ),( 7.8 )and( 7.9 ).Since s rr = 0at r = R and d s rr dr < 0, s rr always positive.Similarly,thederivativeofthetangentialstressis d s qq dr = g E d r dr + g E r 3 Z r 0 d r d z z 2 d z ,(7.18) 137 andtheinvarianceofstresstensor s qq + s rr = g E ( r r 0 )+ 2 g E R 2 Z R 0 ( r r 0 ) z d z .(7.19) NoconclusioncanbemadeaboutthesignofexpressionsinEq.( 7.18 )andEq.( 7.19 )so tangentialstresscouldbeeithertensileorcompressiveassumingthemassislostfromthe center d r dr > 0. 7.2Ananalyticalderivationfortheradialstress Inthisderivation,theshrinkagestrainistakenas # ( r ) whichisageneralfunction ofradius r .StartingfromEq.( 2.7 ),theexpressionsoftotalstrainsforaxialsymmetry conditionincylindricalcoordinatesare # rr = du dr # qq = u r (7.20) GoverningequationorstressbalanceisgivenbyEq.( 7.5 ).SubstitutingEq.( 7.20 )and ( 7.6 )intothegoverningequation,Eq.( 7.5 )gives d dr du dr + n u r 1 + n r # 0 +( 1 n )( du dr u r )= 0.(7.21) Expandingtheaboveexpressiontogetthesecondorderordinarydifferentialequation whichistheEuler'sequation d 2 u dr 2 + 1 r du dr u r 2 =( 1 + n ) d # 0 dr .(7.22) Thegeneralsolutionof( 7.22 )canbeobtainedviausingvariablechange: r = r 0 e t so dr = r 0 e t dt and dt = e t r 1 0 dr = r 1 dr ,where r 0 isanycharacteristiclengthscale.Thus: du dr = du dt dt dr = e t r 1 0 du dt d 2 u dr 2 = d dt e t r 1 0 du dt dt dr = d 2 u dt 2 du dt ! e 2 t r 2 0 138 substitutingtheseexpressionsbackinto( 7.22 )yields ( u 00 t u 0 t + u 0 t u ) e 2 t r 2 0 =( 1 + n ) d # 0 dt e t r 1 0 (7.23) or ( u 0 t + u ) 0 ( u 0 t + u )=( 1 + n ) d # 0 dt e t r 0 .(7.24) with u 0 t = du dt and u 00 t = d 2 u dt 2 .Introducethenewdependentvariable w ( t ) w ( t )= u 0 t + u (7.25) andsubstitutethisinto( 7.23 )aftermultiplying( 7.23 )withtheintegratingfactor e t ( w ( t ) e t ) 0 = e t ( 1 + n ) d # 0 dt e t r 0 (7.26) integratingbothsidesgives w ( t )= e t ( 1 + n ) r 0 Z t 0 d # 0 dt dt + A ,(7.27) whichcanbewrittenas w ( t )= e t ( 1 + n ) r 0 ( # 0 + A ) (7.28) whereAisanintegralconstantwhichcouldbedeterminedlaterfromboundarycondi- tions. u ( t ) canbesolvedfrom w ( t ) ( u ( t ) e t ) 0 = w ( t ) e t = e 2 t ( 1 + n ) r 0 ( # 0 + A ) whichgives u ( t )= e t Z t 0 e 2 t ( 1 + n ) r 0 ( # 0 + A ) dt + B 139 nowchangingbacktonormalvariable r usingtherelation e t = r 0 r 1 u ( r )= r 0 r 1 Z r 0 r 2 r 2 0 ( 1 + n ) r 0 ( # 0 + A ) r 1 dr + B thenthescalinglength r 0 iscancelledoutandthegeneralsolution u ( r ) is u ( r )= r 1 Z r 0 r 2 ( 1 + n ) ( # 0 + A ) r 1 dr + B (7.29) hereA,Baretwoconstantsofintegration,inwhichBmustvanishsince u ( r ) isat r = 0.Thisgives u ( r )= r 1 ( 1 + n ) Z r 0 # 0 ( z ) z d z + A r 2 2 ! ,(7.30) Aisfoundfortwocases,theonehastractionfreeboundaryconditionwhichrequires theradialstress s rr tovanishontheouterradius.From( 7.30 ), du dr = 1 + n r 2 Z r 0 # 0 ( z ) z d z +( 1 + n ) # 0 +( 1 + n ) A 2 .(7.31) Substitute du dr and u ( r ) from( 7.30 )and( 7.31 )intothefollowingexpressionforradialstress s rr = E 1 n 2 du dr + n u r E 1 n # 0 ,(7.32) whichisobtainedfrom( 7.20 )and( 7.6 ),givesthefollowingexpressionforradialstress s rr = E r 2 Z r 0 # 0 ( z ) z d z + E ( 1 + n ) 1 n A 2 .(7.33) Thetractionfreeconditionatouterradius s rr ( r = R )= 0requires E R 2 Z R 0 # 0 ( z ) z d z + E ( 1 + n ) 1 n A 2 = 0(7.34) andthus: A = 2 ( 1 n ) ( 1 + n ) R 2 Z R 0 # 0 ( z ) z d z (7.35) 140 TheparameterAissubstitutedbackintotheexpressionfordisplacement( 7.30 )andstress ( 7.33 ),giving u ( r )=( 1 + n ) 1 r Z r 0 # 0 ( z ) z d z + r R 2 ( 1 n ) ( 1 + n ) Z R 0 # 0 ( z ) z d z (7.36) s rr = E R 2 Z R 0 # 0 ( z ) z d z E r 2 Z r 0 # 0 ( z ) z d z (7.37) andintotheexpressionforthetangentialstress,whichis s qq = E 1 n 2 ( u r + n du dr ) E 1 n e 0 ,(7.38) using( 7.30 )and( 7.31 ),itbecomes: s qq = 1 r 2 Z r 0 # 0 ( z ) d z + E ( 1 + n ) 2 ( 1 n 2 ) A 2 E # 0 (7.39) SubstitutingAin( 7.35 )intotheaboveexpressiongives s qq = E r 2 Z r 0 # 0 ( z ) d z + E R 2 Z R 0 # 0 ( z ) d z E e 0 (7.40) Threeequations( 7.36 ),( 7.37 ),( 7.40 )aretheanalyticalsolutionsfortheaxissymmetric shrinkagestrainproblemwithtractionfreeboundarycondition. Otherboundaryconditionfortheproblemcouldbezerodisplacement(orpinned)at theouterradius u ( r = R )= 0,thenfrom( 7.30 ) A = 2 R 2 Z R 0 # 0 ( z ) z d z (7.41) subsequently,theexpressionsfordisplacementandstressare u ( r )=( 1 + n ) 1 r Z r 0 # 0 ( z ) z d z r R 2 Z R 0 # 0 ( z ) z d z ,(7.42) and: s rr = E r 2 Z r 0 # 0 ( z ) z d z E ( 1 + n ) 1 n 2 R 2 Z R 0 # 0 ( z ) z d z .(7.43) 141 7.3Numericalresultswithoutcracking TheanalyticalexpressionsoftemperatureandstressesEqs.( 7.3 ),( 7.8 )and( 7.9 )in- volveintegrals,thusnumericalintegrationmustbeultilized.Tosimplifythe integralinEq.( 7.3 ),thedimensionlesstemperature q as q = Z ¥ h e s 2 s ds ,(7.44) thenusingchangeofvariable u = 1 s , z = 1 h ,andthus du = 1 s 2 ds .Substituinginto ( 7.44 ), q = Z 0 z e 1/ u 2 u du ,(7.45) orequivalently q = Z z 0 e 1/ u 2 u du ,(7.46) where z = 1 h = 2 p a r t r .Thusthenewintegrationfordoesnotinvolvethelimit andtemperaturecanbeexpressedas: T = T 0 + q 0 2 p k r Z 2 p a r t r 0 e 1/ u 2 u du .(7.47) ThedensityiscalculatedfromEq.( 7.4 )oncetemperaturedistributionisknown.Without cracking,stresscanbegivenbyEqs.( 7.8 )and( 7.9 ).Oncecrackinghappens,those stressequationsarenolongervalid.Inthissection,thefisemiflanalyticalsolutionsfor temperature,densityandstressesarediscussedforonesetofparametersasgivenbelow. 142 R = 2 cm , T 0 = 300 K , a r = 4 10 7 m 2 s 1 , q 0 2 p k r = 500 K , T a = 12000 K , A = 10 8 s 1 , r 0 = 1000 kgm 3 , r c = 300 kgm 3 , n = 0.45, s c / E = 0.012. (7.48) Withtheseparameters,thedistributionofthedimensionlesstemperature q anddensity r arecalculatedusingEqs.Eqs.( 7.3 ), 7.4 andplottedinFigs. 7.1 and 7.2 ,respectively.In thesees,theradialcoordinateisrescaledbythesampleradius R . Thenthedensitycanbeusedtocalculate s rr and s qq fromEqs.Atthelocationnear samplecenterwhere r issmall,theapproximation( 7.13 )canbeusedtoevaluatestresses forabetteraccuracy.TheirdistributionatdifferenttimesareshownbyFigs. 7.3 and 7.4 . Itcanbeseenfromtheseesthat,while s rr isalwaystensile, s qq canbebothtensile andcompressive.Inthisproblem,massislosingfromthecenter, s qq istensilenearthe centerandcompressiveneartheradius. 143 Figure7.1:Thedistributionofthedimensionlesstemperature q atdifferenttimesfrom1 s to600 s inradialheatingcondition. Figure7.2:Thedistributionofthedimensionlessdensity r atdifferenttimesfrom1 s to 600 s inradialheatingcondition. 7.4Numericalresultswithcracking Oncecracksinitiate,theanalyticalexpressionsforstressesarenolongervalid.With thetemperatureanddensitygiveninSection 7.3 bynumericalintegration,thestress 144 Figure7.3:Thedistributionoftheradialstress s rr atdifferenttimesfrom1 s to600 s in radialheatingcondition. Figure7.4:Thedistributionofthetangentialstress s qq atdifferenttimesfrom1 s to600 s inradialheatingcondition. isresolvedusingFEMwithisotropictriangularmeshofsize10 2 cm,thesameas usedinchapter 3 .Fig. 7.5 showsonesetofcrackpatterncorrespondingto s c / E = 0.0833. Cracksinitiatefromthediskcenterwheretheheatisappliedandspreadoutward. Theytendtoadvanceinthedirectionperpendiculartothecharfront. 145 Figure7.5:Themaximumprinciplestress s 1 when s c / E = 0.0833at t =2,10,20,40,80, 100,120,160,180,200,240,280 s . InFig. 7.6 ,thesamecrackpatterninFig. 7.5 isplottedonthebackgroundofthedensity Fig. 7.7 comparesthecrackpatternsoftwovaluesof s c / E =0.05(left)and0.0833(right). Whilebothhavecomparablecracklength,theonecorrespondingtolowervalueof s c / E producescrackpatternthathassmallerspacing. 146 Figure7.6:Theevolutionofthemaximumprinciplestress s 1 alongwiththecracking processwhen s c / E = 0.0833 t =2,10,20,40,80,100,120,160,180,200,240,280 s . 147 Figure7.7:Maximumprinciplestress s 1 (top)anddensity r (bottom)when s c / E = 0.05(left)and0.0833(right)at t =270 s 148 CHAPTER8 CONCLUSIONSANDRECOMMENDATIONFORFUTUREWORK Atheoreticalandnumericalmodelisdevelopedforcrackdevelopmentinsolidsthat undergothermo-chemicaldecomposition.Itisproposedthatmaterialdeveloptensile stressesandfractureduetomassdepletion.Importantmaterialparametersinthisthe- oryincludethermaldiffusivities,ratioofchartovirginmaterialdensity,pre-exponential factor,activationtemperature,ratioofcrackingthresholdtoYoung'smodulus,Poisson's ratioandmasslosscoefNinedimensionlessparametergroups P 1 through P 9 thatcharacterizethemodelaredetermined.Thenumericalsimulationuseslineartri- angleFEMandimplementedinFORTRANprogramminglanguage.Materialdamage ismodelledbyelementextinctionwhichintroducesanothercharacteristiclengthscale whichisthemeshsize.MPI(MessagePassingInterface)isusedtoparallelizethecode, thusmakingverymeshresolutionpossible.Mathematicalmorphologyalgorithms aredevelopedandcodedinMATLAB.Theyareusedtoanalyzethecrackmorphologies producedandcanbealsoappliedtoabroadrangeofnetworkpatterns.Furthermore,par- tialanalyticalsolutionsarederivedforthecirculardomainunderaxis-symmetriccondi- tionalongwithnumericalsimulationsofthecrack.Fromthenumericalsimulations,itis foundthattherearegroupsthatthemorphologyofthecrackpatternswhileoth- ersjustsimplyrescalethecharacteristictimes.Inparticularly,thecompetitionsbetween twogroups P 3 and P 9 thatcharacterizethethermalanisotropyandtensilestrengthofthe materialrespectively,determinetheloop-likeortree-likebehaviorofthecrackpattern. Inthisstudy,modelingofdamageandthuscrackpropagationisdonebydeletingel- ementswhosemaximumprincipalstressexceedsacertaincriticalvalue.Whilethisisa localapproachtomodelcracksthathaveadvantagesofrequiringnospecialtreatmentof cracks,itmaycausemeshdependence(includingbothmeshsizeandmeshorientation) whichisitsmaindrawback.Inthiswork,linearshapefunctionsareusedfordisplace- 149 ments,thusitsspatialderivativesareconstantovereachelement.Soarethecom- putedstressvalues.Ontheotherhand,thetheoreticalstresssurroundingslitcrack tipshasverysteepchange.Thusthelinearityofshapefunctionsexaggeratesthedepen- denceofthestressesonthemeshsize.Fortunately,thankstoMPIandtheMSUHigh PerformanceComputerICER,themeshsizeusedinthissetofcomputationsisvery comparedwithsamplesize.Thispartiallymitigatestheissueofmeshsizedependence. Furthermore,unlikesomenumericalmodelssuchasthespring/springbundlenet- workorcrackbandmodelsthatpurposelyimposestatisticaldistributiononthespring strengthortensilestrength[ 96 ],[ 206 ],wearenotintendedtointroducerandomnessinto ourmodel.Howeverthesomewhatrandombehaviorofcracksisunavoidablebecauseof itsdependenceonmeshorientation.Sincethesequenceofelementsremovedatthelatter stepsdependsontheonesremovedattheprevioussteps,randomnessgrowsascrack develop.Thiscausescrackmorphologytobeunsymmetricalaboutthemiddlevertical lineofthesampledespiteofthesymmetricboundaryconditions.Moreover,inthisstudy, crackmorphologyisanalyzedfromastatisticalviewpointasitisforvarious parametervalues.Thecrackpatternsontheleftandrightpartsofthesamplearesta- tisticallysimilarasseenfromtheappendix,section C .Anddespitechangeinmeshsize, generalbehaviorsofcracks,thosedeterminedbyphysicalmechanisms,areunchanged asdiscussedintheappendixsection D . Thereareremediestomeshdependenceproblem,amongwhichisthenonlocalfor- mulardevelopedbyBazantandOh[ 94 ],forexampletheworks[ 67 ]and[ 204 ],in whichthestressisreplacedbyitsweightedaveragedvalueoveracircleofsomeaction radius R .Theradius R isafunctionoftensilestrengthandfracturetoughness.Inthis approach,damagehappensatlocationaheadofcracktipwheretheweightedaveraged stressexceedsitstensilestrength.Inlocalmodeltodamage,mostlyinthestudyofelasto- plastic,thereareremediesusingsomelocalizationlimiters.Theforemostworkofthis typebelongstoL'Hermite(1952),inwhichtensilestrengthoryieldlimitisproposedtobe 150 dependentonstraingradient,FloeglandMang(1981),SchreyerandChen(1984),Mang andEberhardteiner(1986)straingradientintoyieldfunction;Aifantis(1984),Bazantand Belytschko(1987)higherorderstraingradientisincludedintostrengthIn Abaqus[ 205 ],itisdoneviascalingthestrainwithcharacteristicelementsizeasdiscussed inSection24.2.3flDamageevolutionandelementremovalforductilemetalsfl.[ 204 ]re- ducesthetensilestrengthwhenlargerelementsizeisusedwhile[ 203 ]suggests keepingtheelementsizeintheregionsaroundcracktipstoavoidmeshde- pendence.[ 202 ]studiestheofminimumelementsizeonthedeterminationof stressintensityfactor.Willis[ 207 ]showsthatonepracticethatcanbeusedtoreducethe themeshdependenceofFEMcommercialcodesistoadjustthefractureenergywithcho- senelementsize.AccordingtothestudyofGuo et.al. [ 208 ],thechosenmeshsizeshould besmallenoughcomparewithplasticzonelengthtocapturethegradientsaroundcrack tips.Whenthemeshsizeissufthezaggedsurfacecanrepresentmicrode- fects. Otherthandamagemodels,therearerecentlydevelopednumericalapproachesto crackswhichareshowntobemeshindependent.Suchexamplesincludedamageme- chanics,phasemodelorXFEMwhichareapplicabletovarioustypesofmaterials fromelastictoviscoelasticandplastic.MoredetailsaboutthederivationsoftheXFEM andphasemodelforthelinearelasticfractureproblemarefoundattheAppendix, Sections E and F .Theproblemcanbeextendedto3Dwhichismuchmorecomputational expensivebutisexpectedtobefeasiblewiththeuseofparallellibrarysuchasMPI. Themodelconsideredinthisstudythegreatlyonefromthegeneralroad mapoutlinedearlierintheIntroductionchapter.Thustherearegreatpossibilitiesto expandittoobtainamoredetaileddescriptionsoftherelatedphysicalprocesses.For example,thesolidmaterialisnotnecessaryonlyisotropic.Alsolargedeformationap- proachcanbeusedformaterialsexhibitingnonlinearelasticbehavior.Moreover,ther- maldegradationturnssolidsintoporoussubstances,thusporoelasticitymodelcanbe 151 used.Currentmodelassumesthatmaterialdamageisinducedbymassdepletionstress whicharetensile,amoregeneralmodelcanconsiderdamageduetocompressivestress. Moreoverthermalandmechanicalpropertiessuchasthermaldiffusivities,mechanical strength,Young'smodulus,etc...canchangewithtemperatureanddensity.Inthismodel, crackshavenouponheattransferinsidesolids.Moredetailedmodelcantreat cracksasadiabaticwhichcausestemperaturejumpacrosstheirsurfacesusingthedis- continuouscrackfunctionofXFEM.Theeffectsofmechanicalboundaryconditionsor constraintscanbeconsidered,forexamplegripcanbeusedinsteadofrollercondition. Currentworkhasnotyetmodeltheradiativeandconvectiveheattransferpro- videdbytheisrepresentedbyaconstantheatinstead.Futureworkcanin- cludechemicalreactionsandtransportationofenergyandmomentumofspeciesinthe gasphasetomodelignitionandspread.Thisrequiressolvingsetofchemically reactiveconvectivediffusivewequationsinthegasphase.Reactantsusuallyinvolve hundredsofspecieswhicharefreeradicalandionsbutcanbeusingmixture fractionformulation.TheresultingPDEsystemsforspeciesconcentrationandare highlynonlinearbecausetheyinvolvetheexponentialArrheniusterms,convectiveterms andneedtobesolvedbynonlinearnumericalmethods.Movingexertsshearforce onthesoliduppersurface.Furthermore,asoutlinedearlierintheIntroductionchapter, thereisacouplingbetweencrackofsolidphaseandingasphase.Inparticular,the existenceofcracksallowsgaseoustobereleaseddirectlyintogasphase.High pressurealsoexertsanopeningtractiononcracksurfaces. 152 APPENDICES 153 APPENDIXA ANANALYTICALSOLUTIONUNDERSIMPLIFIEDCONDITIONS. Thestressequilibriumequations,Eqs.( 2.23 ),( 2.24 )canbewrittenintermsofdisplace- ments u and v usingEq.( 2.22 ),whichrelatesthestraintothedisplacementsandEqs. ( 2.19 ),( 2.20 ),( 2.21 ),whichrelatestresstostrainas ¶ 2 u ¶ x 2 + 1 n 2 ¶ 2 u ¶ y 2 + 1 + n 2 ¶ 2 v ¶ x ¶ y = g 1 + n 2 r 0 ¶r ¶ x ,(A.1) ¶ 2 v ¶ y 2 + 1 n 2 ¶ 2 v ¶ x 2 + 1 + n 2 ¶ 2 u ¶ x ¶ y = g 1 + n 2 r 0 ¶r ¶ y .(A.2) Inthespecialcasewhen v = v ( y ) , u = 0,Eq.( A.1 )becomes ¶r ¶ x = 0andEq.( A.2 ) becomes ¶ 2 v ¶ y 2 = g 1 + n 2 r 0 ¶r ¶ y ,(A.3) whichisintegrable.Iftheboundaryconditionsareappropriate,ananalyticalsolution exist.With u = 0, v = v ( y ) ,therollerconditionat y = 0reducesto: y = 0: v = 0,the tractionfreeconditionat y = H becomes ¶ v ¶ y = g 1 + n 2 r r 0 r 0 .(A.4) Nowthequestionis,which2Dversionoftheboundaryconditionsfortheleftandright surfacesdonotcontradictwiththeassumptionthat u = 0, v = v ( y ) .Itturnsoutthatthis correspondstorollerconditions.SolvingEq.( A.3 )subjectto y = 0, v = 0andEq.( A.4 )at y = H leadtothesolution v ( y )= g 1 + n 2 1 r 0 Z y 0 r ( h ) d h y (A.5) providedthat r isnotdependentonthecoordinate x . 154 APPENDIXB NUMERICALRESULTSFORTHECASE s c / E = 0.024. IntheSection 3.4 ,thevalue s c / E = 0.0333isused.Thispartpresentstheresultsfor s c / E = 0.024whileotherparametersarekeptthesameasthoseinSection 3.4 .Fig. B.1 showsthedistributionofthemaximumprinciplestress s 1 andtheevolutionofcracksup to t = 6000 s attimesindicatedintheecaption. Basedontheevolutionoftemperature,densityandcrackmorphology,theprocess canbedividedintofourstages,whichare:(a)inertheating(b)pyrolysis,notcharred and,crackinitiation,(c)slightlycharred,mainlyinitiation,(d)halfcharredand fastpropagation,(e)almostcharred,deceleratedpropagation.Inthestage(a),the sampletemperaturecontinuesrisingbutisstilltoosmalltoproduceanappreciablemass Thisissameasthestageof[ 11 ].Theevidenceforthisstageisbasedonvalues ofthesurfacedensityinwhich r = 0.9995happenningat t = 18 s isused.Thestage thenistakentoevolvefrom t = 0to t = 18 s .Followingthisstage,instage(b),pyrolysis beginsbutthesampledensityisstillwellabovethecharvalue.Thecrackingthreshold hasnotyetbeenattainedbythemaximumprinciplestressduetoaninsufdensity gradient.Thedensitygradienteventuallyattainsasufhighvalueforthe cracktonucleateat t = 68 s ,whichmarkstheendofstage(b).Atthistime,thelowest densityvalueis r = 0.8345. Thethirdstage(c)ischaracterizedbyadensitywhosegradientdecaysas t 1/2 andcon- tinuetowardthenextstage,asmentionedintheprevioussection.cracksinitiatefrom theheatedsurfaceandpropagateradiallydownward.Thedensityattainsthechar value r = 0.3when t = 116 s .Eventually,initiationactivityisdiminishedandreplaced bycrackelongationandbranching,whichindicatestransitiontothenextstage,stage(d) .Attheendofstage(d)thedensityatthelowersurfacealsoattainsthecharvalue.Crack initiationisnolongerobservedandtheexistingcrackspropagateataslowerpace.These 155 FigureB.1:Maximumprinciplestress s 1 .Plotsfromlefttoright,toptobottom,corre- spondto t =50,100,300,700,1000,1500,2000,2500,3000,4000,4500,5000,5700,6000 s , respectively 156 cracksnowintersecteachother,formingloopsandnetwork-likepatterns.The timesofeachstageisthefollowing:(a)from0 s to18 s ,(b)from19 s to68 s ,(c)from68 s to 116 s ,(d)from117 s toabout4500 s ,(e)fromabout4500 s onward. FigureB.2: q attheendofthestage t = 18 s when r = 0.995atthemiddleofthe heatedregionontheuppersurface. Thecorrelationbetweencrackpropagationandthedensityisseenmoreeasily byplottingcracksoverthedensityseeFig. B.3 .Itisnotedthatinthismodel,the evolutionofdensityaswellastemperatureisnotaffectedbycrackpresence. Itcanbeseenfromplotsforearlytimesthatthecrackpatternexhibitshierarchical behaviorsuchthatlongandshortbranchesalternatewitheachotherduringthisstage. Thiselegantpatternissimilartothedistinctiveperiodicdoublingpatternrecognizedin quenchingorcoldshockexperimentsby[ 67 ]inrectangleorin[ 126 ]circularsample. B.4 showshowcrackpatternsareaffectedbygroup P 9 and P 3 157 FigureB.3:Plotsof r insamplewithcrackingpatternat t =100,300,1000,2000,3000,4500, 5700,6000 s .At t = 100 s ,crackshavealreadydevelopedwhiletheuppersurfacehasnot beencharredyet. FigureB.4:Left:Loopbehaviordependson P 9 = s c / E .Right:Crackpatternchange with P 3 = a y / a x . 158 APPENDIXC AVERAGECRACKSPACINGONTWOHALVESOFTHESAMPLE AsmentionedinSection 3.4 ,thecrackpatternsontheleft( x < 0.5)andontheright ( x > 0.5)ofthemidverticallinearenotexactlythesameinspiteofsymmetricboundary conditions.However,thetwosidesarestatisticallysimilarsincethenumberofbranches oraveragespacingontwosidesareclose.ItcanbeseenfromFig. C.1 thatcompares theaveragespacingofthewholesamplewiththeaveragespacingofcracksontheleft side.Twosamplesoflowercrackingthreshold, s c / E = 0.024(upper)and s c / E = 0.033 (lower),arechosenbecausetheyproduceahighernumberofcracksforbetterstatistics. FigureC.1:Averagecrackspacingofthewholesampleandaveragecrackspacingonthe lefthalfofthesample( x < 0.5)fortwocasesofcrackingthreshold s c / E = 0.024(upper) s c / E = 0.033(lower). 159 APPENDIXD CASE s c / E = 0.033 USINGDIFFERENTMESHSIZES Inthesesetsofsimulation,themeshsizeischangedtoexamineitseffectsoncrackmor- phology.Inparticular,mesh h e /2andcoarsermesh2 h e areusedinthissection, comparedtotheoriginalmesh h e usedinthemainpartofthethesis. Figure D.1 and D.3 showsthecrackpatternsatdifferenttimesfor s c / E = 0.033using2 h e and h e /2mesh,respectively.ItcanbeseenbycomparingthreecasesinFigures D.1 (2 h e ), D.2 ( h e )and D.3 ( h e /2)thatdespitethedifferenceinthelocaldetails,thegeneralevolution behaviorsrelatedtoinherentphysicalmechanismsremainunchanged.More, cracksopenupinthedirectionsperpendiculartothepyrolysisfrontorcontourlevelof densityandadvanceintocharregion;theintersectionsaremadeatanrightangle; hierarchicalpatternsatearlystage;cracksavoidcoldwalls. FigureD.1:Maximumprincipalstress s 1 / E correspondingto s c / E = 0.033at t =100, 1000,3000and5000 s fromtoptobottom,respectively.Meshsize2 h e isusedtoproduce thissetofsimulation. Thetotalnumberofcrackbranchesorequivalently,theaveragecrackspacing g over threemeshes2 h e , h e and h e /2arecomparedbyplottingthemonthesameeasshown byFig. D.4 .Itcanbeseenthattheaveragecrackspacingforallthreemeshesarecom- parable.Moreover,thecrackpatternsofthemesh( h e /2)havelesschangethanthe 160 FigureD.2:Maximumprincipalstress s 1 / E correspondingto s c / E = 0.033at t =100, 1000,3000and5000 s ,fromtoptobottom,respectively.Meshsize h e isusedtoproduce thissetofsimulation.NotethatthisisthesamecrackpatternasshownbyFig. 3.8 FigureD.3:Maximumprincipalstress s 1 / E correspondingto s c / E = 0.033at t =100, 1000,3000and5000 s ,fromtoptobottom,respectively.Theelementsizeusedto producethissetofcracksis h e /2. coarsermesh(2 h e )whencomparingwiththeoriginalmesh( h e ).Thestressgradientson h e and h e /2areresolvedbecausethemeshesaresufStressesdecaystothe valuezeroawayfromcracktipsoveradistanceofapproximatelytenelementsin thesetwomeshes.Thuswhilethemeshsizecannotbechosenarbitrarily,andmeshde- pendencycannotbetotallyeliminated,thevalue h e usedhereissufsmallsothat reductiondoesnotleadtodifferentcrackpatterns.Moreover,whenus- ingelementremoval,meshsizeshouldbeinterpretedasaninherentlengthscaleinthis problemandshouldbechoseninrangethatissuitabletosamplesizeanddefectsize. 161 FigureD.4:Averagecrackspacingusingthreemeshes2 h e , h e and h e /2for s c / E = 0.033 162 APPENDIXE ANXFEMFORMULARFORTHETHERMOELASTICFRACTUREPROBLEM TheXFEMdiscretizationpresentedin[ 99 ]forthethermoelasticproblemcanbeapplied directlytoourstudyinwhichtheshrinkagecausedbytemperaturedifferenceisreplaced byshrinkagecausedbymassdepletion.Consideranelasticbodyinadomain W with crack G ,subjectedtoanexternalbodyforce ~ X andthefollowingboundaryconditionsfor displacementsandstresses u = uon G u s . n = ton G t (E.1) s . n = 0 on G c Whichareprescribeddisplacementon G u ,prescribedtractionon G t andtractionfreeon G c .Thefollowingequationsofmotion,equationofenergyandconstitutiveequationsare adaptedfrom[ 99 ] r ¶ 2 u i ¶ t 2 = ¶s ij ¶ x j + X i (E.2) and ¶ T ¶ t = a ¶ ¶ x i ¶ T ¶ x i + R .(E.3) thegeneralconstitutiverelationsforthemassdepletionorthermoelasticstressaretaken as s ij = 2 m# ij + l# kk d ij + bcd ij (E.4) wherethecoef b istakentobe g formassdepletionor g t ( 3 l + 2 m ) forthermoe- lasticproblemandthefunction c is r 1or T T 0 respectively.Theboundaryconditions 163 fortheheatconductionproblemaretakenasprescribedheat q 0 ontheboundary G q . TheXFEMinvolvesenrichingthenodessurroundingcracksbytheHeavisidefunc- tionsandthecracktipsfunctionsgivenbyEq.( 1.11 ).Eachcrackcanbetrackedbyset ofsegments(orpoints,equivalently)thatarenotgridnodalpointsorbylevelset.This methodcanalsobeappliedtomultiplecracksituations[ 98 ].Asacriterionforadvance- mentofthedirectionofthecracktip,theangle q canbebasedeitheronenergyorstress aroundcracktip.Crackadvancesinthedirectionthatmaximizestherateofstrain energyrelease,whichisalsothedirectionofmaximumtangentialstress.Thedisplace- mentsareapproximatedas: ~ u ( x , y )= å n 2 N N n ( x , y ) ~ a n + å n 2 N cr N N ( x , y )[ H ( x , y ) H ( x n , y n )] ~ b n + å n 2 N tip N N ( x , y ) M å m = 1 [ F m ( r , q ) F m ( r n , q n )] ~ c mn (E.5) where N isthetotalnumberofnodes, N n ( x , y ) areusualshapefunctions(Lagrangetype), F m ( r , q ) , m = 1,4areshapefunctionsgivenby( 1.11 ), N c r isthesetofnodeswhosesup- portiscutbythecrackand N tip isthesetofnodessurroundingthecracktip.Inthis study,temperatureistakentobediscontinousacrossthecrackface.Basedontheanalyt- icalderivationoftheasymptotictemperaturesolutionnearthecracktip,thetemperature isdiscretizedas: T ( x , y )= å n 2 N N n ( x , y ) a n + å n 2 N cr N N ( x , y )[ H ( x , y ) H ( x n , y n )] b n + å n 2 N tip N N ( x , y )[ F 1 ( r , q ) F 1 ( r n , q n )] c n (E.6) Thematricesobtained,includingmass,stiffnessandforcematrices,derivedby[ 99 ], 164 aresimilartothoseusingtheFEMexceptstherearepartsinvolvingtheenrichedfunctions y (includingbothHeavisideandcracktipsfunctions). [ C q ]= 2 6 4 [ C aa ][ C ad ] [ C ad ][ C dd ] 3 7 5 (E.7) [ K q ]= 2 6 4 [ K aa ][ K ad ] [ K ad ][ K dd ] 3 7 5 (E.8) F q = 2 6 4 ( F a ) ( F d ) 3 7 5 (E.9) wherethe ij componentsofthematrices [ C aa ] , [ C ad ] , [ C dd ] aretakenas C aa ij = R W N i N j d W , C ad ij = R W N i Y j d W , (E.10) C dd ij = R W Y i Y j d W . andthecomponentsofthe K matricesare K aa ij = R W a ¶ N i ¶ x ¶ N j ¶ x + ¶ N i ¶ y ¶ N j ¶ y ! d W , K ad ij = R W a ¶ N i ¶ x ¶ Y j ¶ x + ¶ N i ¶ y ¶ Y j ¶ y ! d W , (E.11) K dd ij = R W a ¶ Y i ¶ x ¶ Y j ¶ x + ¶ Y i ¶ y ¶ Y j ¶ y ! d W . the i componentsoftheforcesforthethermalproblemare 165 F a i = Z W N i Rd W Z G q N i q 0 d G ,(E.12) and F d i = Z W Y i Rd W Z G q Y i q 0 d G .(E.13) Fortheelasticproblem,letthevectorofunknownbe ( u i , v i , d u i , d v i ) ,where u and v arethe horizontalandverticaldisplacements, ( u , v ) T = ~ a , d u i , d v i arethe x and y componentsof theenrichmentdegreeoffreedom ~ d thatincludestheHeavisidecrackfaceparts ~ b andthe stressfunctioncracktippart ~ c .Thematicesoftheelasticproblemarethefollowing [ M e ]= 2 6 6 6 6 6 6 6 4 [ M aa ][ 0 ][ M ad ][ 0 ] [ M aa ][ 0 ][ M ad ] [ M dd ][ 0 ] Sym . [ M dd ] 3 7 7 7 7 7 7 7 5 (E.14) [ K e ]= 2 6 6 6 6 6 6 6 4 [ K aa xx ][ K aa xy ][ K ad xx ][ K ad xy ] [ K aa yy ][ K ad yx ][ K ad yy ] [ K dd xx ][ K dd xy ] Sym . [ K dd yy ] 3 7 7 7 7 7 7 7 5 (E.15) [ K e ]= 2 6 6 6 6 6 6 6 4 ( F a x ) ( F a y ) ( F d x ) ( F d y ) 3 7 7 7 7 7 7 7 5 (E.16) 166 M aa ij = Z W r N i N j d W M ab ij = Z W r N i F j d W (E.17) M bb ij = Z W r F i F j d W K aa xx , ij = Z W ( 2 m + l ) ¶ N i ¶ x ¶ N j ¶ x + m ¶ N i ¶ y ¶ N j ¶ y ! d W K aa xy , ij = Z W m ¶ N i ¶ y ¶ N j ¶ x + l ¶ N i ¶ x ¶ N j ¶ y ! d W K aa yy , ij = Z W m ¶ N i ¶ x ¶ N j ¶ x +( 2 m + l ) ¶ N i ¶ y ¶ N j ¶ y ! d W K ab xx , ij = Z W ( 2 m + l ) ¶ N i ¶ x ¶ F j ¶ x + m ¶ N i ¶ y ¶ F j ¶ y ! d W (E.18) K ab xy , ij = Z W m ¶ N i ¶ y ¶ F j ¶ x + l ¶ N i ¶ x ¶ F j ¶ y ! d W K ab yx , ij = Z W m ¶ N i ¶ x ¶ F j ¶ y + l ¶ N i ¶ y ¶ F j ¶ x ! d W K bb xx , ij = Z W ( 2 m + l ) ¶ F i ¶ x ¶ F j ¶ x + m ¶ F i ¶ y ¶ F j ¶ y ! d W K bb xy , ij = Z W m ¶ F i ¶ y ¶ F j ¶ x + l ¶ F i ¶ x ¶ F j ¶ y ! d W K bb yy , ij = Z W m ¶ F i ¶ x ¶ F j ¶ x +( 2 m + l ) ¶ F i ¶ y ¶ F j ¶ y ! d W 167 F a x , i = Z W N i X 1 d W + Z G t N i t 1 d G + Z W ¶ N i ¶ x g ( r 1 ) d W F a y , i = Z W N i X 2 d W + Z G t N i t 2 d G + Z W ¶ N i ¶ y g ( r 1 ) d W (E.19) F d x , i = Z W F i X 1 d W + Z G t F i t 1 d G + Z W ¶ F i ¶ x g ( r 1 ) d W F d y , i = Z W F i X 2 d W + Z G t F i t 2 d G + Z W ¶ F i ¶ y g ( r 1 ) d W ThetimeintegrationcanbetreatedusingtheCrank-Nicholsonschemefortransientheat conductionortheNewmarkmethodfortheunsteadystressbalanceequation.Integration canbeevaluatednumericallyusingGaussianquadratureandLagrangebasisfunctions. Whilethecomputationalmeshisnotrnedfutherwhencracksdevelop,elementsthat arecutbycrackorcontaincracktipsaredividedfortheintegrationpurpose.Thedetailed descriptionoftheintegrationprocesscanbefoundat[ 99 ].Thecrackscanbedescribed explicitlyorimplicitlyusinglevelsetmethodwithtwounsignedlevelsetfunctionsand onesignedlevelsetfunction.Inthelatterapproach,thelocalpolarcoordinateandthe Heavisidefunctioncanbeconstructedfromthosethreelevelsetfunctions. Thedirectionforcrackadvancementisdeterminedfromthemaximumcircumferen- tialstresscriteria.Otherstudiesusedifferentcrackadvancementcriterion,suchasthe maximumstrainenergyreleaserate,materialforceorminimalstrainenergydensity.Ac- cordingtomaximumcircumferentialstresscriteriawhichisalocalapproachbasedon stressldnearcracktip,crackextendsinthedirection q m thatmaximizesthetangential stress s qq whichisdependendonthestressintensityfactors s qq = 1 p 2 p r K I 4 3 cos q 3 + cos 3 q 2 + K II 4 3 sin q 2 3 sin 3 q 2 ,(E.20) 168 thecondition ¶s qq ¶q = 0at q = q m leadsto K I sin q m + K II ( 3 cos q m 1 )= 0.(E.21) TheimplementationofXFEMonAbaqusbyUsersubroutines(UEL)isalsodoneby otherstudies,suchas[ 101 ]and[ 102 ]withsuccess.XFEM,however,likeFractureMe- chanics,isunabletopredictcrackinitiation.Thisdrawbackisovercomebyotherfracture modelssuchasphasemodel,gradientdamagemodelorcontinuumdamagemodel. 169 APPENDIXF AVARIATIONALFORMULAFORTHEPHASEFIELDMODELOFFRACTURE ConsidertheelasticityproblemdescribedbyEq.( E.1 ).Sincetheconstitutiverelations betweenstressandstrainisgivenbyEq.( E.4 ),theelasticstrainenergydensityis y e ( # )= 1 2 l# ii # jj + m# ij # jj + 1 2 bc# ii .(F.1) Thetotalpotentialenergyofthebodyequaltothesummationofelasticenergyandfrac- tureenergy y p ( u )= Z W y e ( # ) d W + Z G G c ds (F.2) where G c isthefractureenergyperunitcrackarea.Ifsteady-stateassumptionistaken, theLagragianofthesystemwillincludeakinetictermwhichis y k ( u )= 1 2 Z W r ¶ u i ¶ t ¶ u i ¶ t d W (F.3) Forquasi-steadystatecondition,thekineticpotentialisnegligibleintheLagragian.The potentialbyexternalforceis y f ( X , t )= Z W X i u i d W + Z G t t i u i ds (F.4) NowturningtothevariationalformulaoffractureusingthephaseInphase model,thecrackisdescribedassmearedoveralength l c byascalarvariable c thatcan getanyvaluebetweenunityandzero.In1D, c canbeassumedoftakingtheexponential formas c ( x )= exp x x 0 l c (F.5) 170 inwhich x 0 thelocationofthecrackwherereas c getsthevalueofunity.Faraway from x 0 , c dropstozerovalue.Obviously, c ( x ) isthesolutionofthefollowingdifferential equation c l 2 c d 2 c dx 2 = 0(F.6) subjectedtoboundaryconditions c ( ¥ )= 0.Followingthevariationalprinciple,the weakformof( F.6 )is ¥ Z ¥ dv dx dc dx + vc l 2 c dx = 0(F.7) andthefunctionalassociatedwiththeweakformis I ( c )= 1 2 ¥ Z ¥ dc dx 2 + c 2 l 2 c dx .(F.8) Thustheexpressionof I ( c ) in2Disgivenby I ( c )= 1 2 ZZ W r c . r c + c 2 l 2 c dxdy .(F.9) Thefracturesurfaceenergydensityisintermofthephasefunction c ( x , y ) as g ( c )= 1 2 " l c r c . r c + c 2 l c # (F.10) From( F.10 )and( F.2 ),thetotalpotentialenergyofthesystemincludethestrain energyandthecracksurfaceenergywhichis y ( u , c )= ZZ W g ( c ) y e ( # ) d W + G c 2 ZZ W " l c r c . r c + c 2 l c # d W (F.11) inwhich y e isgivenbyEq.( F.1 ), g ( c ) issomedegradationfunction,takenby[ 103 ]as g ( c )=[( 1 c ) 2 + d ] (F.12) 171 d isasmallpositivenumberfornumericalstability.From( F.11 ),( F.4 )andprincipalof virtualdisplacement,thegoverningequationsarederived,whichare [( 1 c ) 2 + d ] ¶s ij ¶ x j + X j = 0 in W [( 1 c ) 2 + d ] n j s ij = t i on G t u i = u i on G u (F.13) G c l c ¶ c ¶ x i ¶ c ¶ x i + G c l c + 2 y e ( # ) = 2 y e ( # ) 2 W ¶ c ¶ x i n i = 0 on G ThestiffnessandresidualmatricesoftheFEMformulationarederivedforthephase variable c andthedisplacement u in[ 104 ]. 172 BIBLIOGRAPHY 173 BIBLIOGRAPHY [1] Parker,W.J.Predictionoftheheatreleaserateofwood. 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