#RACK0ROPAGATIONOF-ETAL0OWDER#OMPACT
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3IMULATIONOFCRACKPROPAGATIONINMETALPOWDERDURINGTHECOLDCOMPACTIONPROCESSISPRESENTEDIN THISPAPER"ASEDONAFRACTURECRITERIONOFGRANULARMATERIALSINCOMPRESSIONADISPLACEMENTBASEDlNITE ELEMENTMODELHASBEENDEVELOPEDTOSIMULATETHEFRACTUREPROCESSINAMULTILEVELCOMPONENTMADE OFIRONPOWDER%STIMATIONOFFRACTURETOUGHNESSVARIATIONWITHRELATIVEDENSITYISESTABLISHEDINORDERTO PROVIDETHEFRACTUREPARAMETERSASCOMPACTIONPROCEEDS!lNITEELEMENTMODELWITHADAPTIVEREMESHING TECHNIQUEISUSEDTOACCOMMODATECHANGESINGEOMETRYDURINGTHECOMPACTIONANDFRACTUREPROCESS WHILEFRICTIONBETWEENCRACKFACESISMODELLEDUSINGTHESIXNODESISOPARAMETRICINTERFACEELEMENTS 4WOWIDELYUSEDYIELDCRITERIAFORPOWDERCOMPACTNAMELY-OHR#OULOMBAND%LLIPTICALCAPAREUSED INTHEMODELS$IFFERENTCRACKGROWTHPATTERNSOBTAINEDBYUSINGTHESETWOYIELDCRITERIAAREPRESENTED ANDCOMPAREDINTERMSOFTHEINmUENCEOFSHEARSTRESSANDRELATIVEDENSITYDISTRIBUTIONS%VENTHOUGH THECRACKSTARTSATDIFFERENTCOMPACTIONSTEPANDDIFFERENTCRACKPATTERNSAREOBTAINEDWHENDIFFERENT YIELDCRITERIAISUSEDSHEARCRACKISPREDICTEDTOSTARTSINTHEREGIONWITHLOWERRELATIVEDENSITYANDHIGHER SHEARSTRESSINBOTHMODELS
+EYWORDS0OWDERCOMPACTFRACTURECRITERIAFRACTURETOUGHNESSlNITEELEMENT
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0ENYELAKUANPERAMBATANRETAKSEMASAPROSESPEMADATANSERBUKLOGAMDIBENTANGKANDIDALAMKERTAS INI"ERDASARKANKRITERIARETAKBAGIBAHANSERBUKTERMAMPATMODELUNSURTERHINGGABERDASARKANANJAKAN TELAHDIBANGUNKANUNTUKMENYELAKUPROSESKERETAKANDALAMKOMPONENBERBILANGARASYANGDIHASILKAN DARIPADASERBUKBESI!NGGARANHUBUNGANKELIATANRETAKDENGANKETUMPATANRELATIFDIBANGUNKANSEBAGAI PARAMETERRETAKSEMASAPROSESPEMADATANDIJALANKAN-ODELUNSURTERHINGGADENGANTEKNIKPENJARINGAN ADAPTIFDIGUNAKANBAGIMENANGANIMASALAHPERUBAHANGEOMETRIYANGBESARSEMASAPROSESPEMADATANDAN KERETAKANMANAKALAGESERANANTARAMUKARETAKDIMODELKANDENGANMENGGUNAKANUNSURANTARAMUKAENAM NODSEPARAMETER$UAJENISKRITERIAALAHYANGBIASADIGUNAKANBAGISERBUKTERPADATIAITU-OHR#OULOMB DANTUKUPELIPSTELAHDIGUNAKAN#ORAKPERAMBATANRETAKBERBEZAYANGDIPEROLEHIDENGANMENGGUNAKAN KEDUADUAJENISKRITERIAALAHTERSEBUTTELAHDIBENTANGKANDANDIBANDINGKANDARISEGIPENGARUHTABURAN
TEGASANRICIHDANKETUMPATANRELATIF-ESKIPUNRETAKBERMULAPADALANGKAHPEMADATANYANGBERBEZA DANCORAKPEMADATANYANGBERBEZADIPEROLEHIAPABILAKRITERIAALAHYANGBERBEZADIGUNAKANRETAKRICIH DIRAMALKANBERMULADIKAWASANYANGBERKETUMPATANRENDAHDANMEMPUNYAITEGASANRICIHYANGTINGGI BAGIKEDUADUAJENISMODEL
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-ANUFACTURING PARTS USING POWDER METALLURGY 0- INVOLVES FOUR MAJOR STEPS POWDER AND LUBRICANT MIXING COMPACTING POWDERS INTO APPROPRIATE SHAPES IN CLOSED DIES TO PRODUCE GREENCOMPACTSSINTERINGTHEGREENCOMPACTS AT ELEVATED TEMPERATURE AND FINALLY POST SINTERINGSECONDARYOPERATIONS#HTOUROUETAL $ETAILSONCOLDCOMPACTIONPROCESSCAN BEFOUNDIN!RIFlN WHERETHENUMERICAL MODELLING OF THE COMPLETE CYCLE HAS BEEN DEVELOPED AND VALIDATED BY EXPERIMENTS )N MODELLINGTHECOMPACTIONPROCESSTHEMACRO MECHANICALMODELLINGAPPROACHISUSEDINTHIS
&)'52%'EOMETRYANDBOUNDARYCONDITIONSOFAROTATIONALmANGEDCOMPONENT Powder
Die
Top punch
Bottom punch
Core rod r = 6.3mm
4.6 mm 15.6 mm
11.7 mm
13.7 mm
WORKWHERETHEPOWDERMEDIUMISCONSIDERED AS A CONTINUUM THAT UNDERGOES LARGE ELASTIC PLASTICDEFORMATION#ONSTITUTIVEMODELBASEDON GRANULARMATERIALISUSEDSINCEPOWDERBEHAVES SIMILARLY TO A FRICTIONAL GRANULAR MATERIAL WITH REGARDTODILATANCYANDDENSIlCATIONBEHAVIOUR 'OLLIONETAL
%XTENSIVE LITERATURE REVIEWS ON MATERIALS UNDER COMPRESSION REVEAL THAT GENERALLY CRACK GROWSINMODE))NOMATTERWHETHERTHEMATERIAL ISBRITTLEORDUCTILEASREPORTEDBY!RUN2OYET AL )SAKSSON AND 3TAHLE
$E"REMAECKERAND&ERIS (OWEVERCRACK PATTERNSAREBEINGINmUENCEDBYTHEAMOUNTOF
APPLIED STRESS AND FRICTION BETWEEN THE CRACK SURFACES)NTHISPAPERASUITABLEFRACTURECRITERION FORMETALPOWDERDURINGCOLDCOMPACTIONPROCESS ISOUTLINEDTAKINGINTOACCOUNTSTHEMECHANICAL BEHAVIOUROFMETALPOWDERUNDERCOMPACTION PROCESS &INITE ELEMENT MODELLING OF THE CRACK INITIATIONANDPROPAGATIONHASBEENDEVELOPED USING&/242!.PROGRAMMINGLANGUAGEWHERE TWODIFFERENTYIELDCRITERIANAMELY-OHR#OULOMB ANDELLIPTICALCAP2OSCOE"URLAND YIELD CRITERIA WHICH ARE ORIGINALLY DEVELOPED FOR SOIL MECHANICSAREUSEDFORPOWDERMATERIAL
&)'52%3TRESSCOMPONENTATAPOINTNEARACRACKTIPINTHEPOLARCOORDINATESYSTEM Arbitrary
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!MULTILEVELCOMPONENTINTHISCASEAROTATIONAL FLANGED COMPONENT IS MODELLED BY AN AXISYMMETRICREPRESENTATIONASSHOWNIN&IGURES AND)RONPOWDERWITHMATERIALPROPERTIESAS LISTEDIN!IDAH ISCOMPACTEDBYBOTTOM AND TO PUNCH MOVEMENTS4OTAL DISPLACEMENT OFTHEBOTTOMPUNCHDBMMWHILETHETOP PUNCHDTMMATTHEENDOFCOMPACTION PROCESS
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)NTHISWORKTHECOMPACTIONISPERFORMEDIN STEPSMOVEMENTOFBOTTOMANDTOPPUNCH RESPECTIVELYANDINTURNASSHOWNIN&IGURE4HIS MEANTHATATOTALDISPLACEMENTOFMMISlRST ACHIEVEDWHENTHEBOTTOMPUNCHHADlNISHEDA STEPSMOVEMENTSTEPTO FOLLOWEDBYA TOTALDISPLACEMENTOFMMBYTHETOPPUNCH AFTERASTEPSMOVEMENTSTEPTO
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%VENTHOUGHITISBELIEVEDTHATFAILUREINMETAL POWDER COMPACTION IS DUE TO SHEAR FRACTURE MODE )) THE FRACTURE CRITERION IN NEED MUST NOT NEGLECT THE POSSIBILITY OF FRACTURE DUE TO OPENINGMODEMODE) 3INCEPOWDERBEHAVE SIMILARLY TO FRICTIONAL GRANULAR MATERIALS LIKE ROCKINCOMPRESSIONFRACTURECRITERIABY1UIHUA ET AL IS ADOPTED IN THIS WORK "ASED ON THE EXAMINATION OF MODE ) AND MODE )) STRESS INTENSITYFACTORSONTHEARBITRARYPLANEθBASED ONTHESTRESSCOMPONENTNEARACRACKTIPASIN
&IGURE +)θ AND +))θ VARYING WITHθO
≤ θ ≤O NO MATTER WHAT KIND OF LOADING CONDITION IS APPLIED THE FRACTURE CRITERION 1UIHUAETAL STATEDTHATFORMODE)FRACTURE TOOCCUR
1<K
K <K
K ,K = K at
II max I max
IIc Ic
I max Ic QIc
&ORMODE))FRACTURE
)N ORDER TO DETERMINE +) MAX AND +)) MAX IN EQUATIONS AND MODE)ANDMODE))STRESS INTENSITYFACTORSINTHEDIRECTIONOFθISDElNED AS
KII( )Q KIsin cosQ Q KIIcosQ sin Q
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+) θ AND +)) θ ARE CALCULATED ON THE MID NODESAROUNDTHECRACKTIPASSHOWNIN&IGURE TOOBTAINTHEMAXIMUM+)MAXAND+))MAXWHERE θ IS DEFINED AS POSITIVE IN THE ANTICLOCKWISE DIRECTIONFROMTHEORIGINALCRACKPLANE+)AND+)) INEQUATIONS AND ARETHESTRESSINTENSITY FACTORS IN THE ORIGINAL CRACK PLANE )N FINITE ELEMENTMETHODTHESEVALUESCANBECALCULATED USING THE DISPLACEMENT CORRELATION TECHNIQUE 0HANETAL ASBELOW
K G
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WHERE'ISTHESHEARMODULUS,ISTHEELEMENT LENGTHκ IS DEFINED ASκnV FOR PLANE STRAINORAXISYMMETRICPROBLEMWHEREVISTHE 0OISSONSRATIO4HEUANDVARETHEDISPLACEMENT COMPONENTS IN X AND Y DIRECTIONS RESPECTIVELY WHERETHESUBSCRIPTSINDICATETHEIRPOSITIONSAS SHOWNIN&IGURE
!$!04)6%-%3(!.$#2!#+-%#(!.)3-
!N ADAPTIVE FINITE ELEMENT MESH USING ERROR ESTIMATORBASEDONSTRESSERRORNORM:IENKIEWICZ :HU ISUSEDWHEREAUTOMATICREMESHING ISCALCULATEDATEACHSTEPDURINGTHECOMPACTION PROCESS %QUATIONS AND ARE USED TO CALCULATETHESTRESSINTENSITYFACTOR3)& OFMODE )+) ANDMODE))+)) 4HESEVALUESARECALCULATED ONTHENODESAROUNDTHECRACKTIPANDCRACKIS MODELLEDTOPROPAGATEINTERELEMENT"OUCHARD ETAL BYSPLITTINGMECHANISMOFCRACKTIP NODEINORDERTOPROVIDETWOADJACENTCRACKFACES 4HIS IS SIMILAR TO THE RELEASE NODE MECHANISM EXCEPTTHATTHEDIRECTIONOFCRACKPROPAGATION ISDEPENDINGONDIRECTIONCRITERIA
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#OULOMBFRICTIONLAWWHICHISOFTENADOPTEDIN FRICTIONPROBLEMSISUSEDINTHISWORKGIVENBY
&)'52%6ARIATIONOFGREENSTRENGTHWITHRELATIVEDENSITY 35
30
25
20
15
10
5
0
0.4 0.45 0.5 0.55 0.6 0.65 0.7 Relative Density, R´
Green Strength,Sus (MPa) Sus =148.66R´ – 65.627
Ff
T S, n T MSNWHEREτ IS THE FRICTION SHEAR STRESSσ. IS THE NORMALSTRESSWHICHSHOULDBECOMPRESSIVEFOR FRICTIONTODEVELOPEDANDμISTHECOEFlCIENTOF FRICTION3IXNODESISOPARAMETRICELEMENTS!RIFlN AREUSEDASINTERFACEELEMENTSFORFRICTION BETWEENPOWDERMATERIALANDDIEWALLDURINGTHE COMPACTIONPROCESSASWELLASTOMODELFRICTION ONTHECRACKFACESINCONTACT
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"ASEDONAPPROXIMATEFORMULASFORMODE)FRACTURE TOUGHNESS+)# BY'IBSONAND!SHBY lNITE ELEMENTMODELSHAVEBEENDEVELOPEDBY#HOI AND 3ANKAR TO ESTIMATE THE VARIATION OF +)# AS A FUNCTION OF SOLIDITY OF CARBON FOAM "Y ASSUMINGTHATTHECRACKISPARALLELTOONEOFTHE PRINCIPALMATERIALAXES+)#CANBEESTIMATEDUSING THEFOLLOWINGFORMULA
WHEREσUS IS THE STRENGTH OF THE MATERIAL AND σMAX IS THE MAXIMUM PRINCIPAL STRESS 3TUDY ON MECHANICAL PROPERTIES OF IRON COMPACT HASBEENPERFORMEDBY0OQUILLONETAL WHICH PROVIDE THE VARIATION OF GREEN STRENGTH WITHRELATIVEDENSITYASSHOWNIN&IGURE5SING
ORENERGYREQUIREDTOCREATENEWCRACKSURFACES 4HUSITISALSOEQUALTOTHEAREAUNDERSTRESSSTRAIN CURVEUPTOFRACTURE%XPERIMENTALDATABY!IDAH ISUSEDINTHISWORKTOOBTAINTHESHEAR STRESSSTRAINCURVESATDIFFERENTCOMPACTIONLOAD ASIN&IGUREANDSUBSEQUENTLYUSEDTOESTIMATE THEVARIATIONOFMODE))FRACTURETOUGHNESS+))#
WITH RELATIVE DENSITY AS DEPICTED IN &IGURE GIVENBY
KIIC 0 1174. e6 5031. R' THE VARIATION OF GREEN STRENGTH EQUATION
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max
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25
20
15
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Shear Stress (MPa)
25 kN 18 kN 15 kN 11 kN 7 kN
&)'52%6ARIATIONOF+))#WITHRELATIVEDENSITY Relative Density, R´
KIIC (MPa m1/2)
KIIC = 0,1174e6,5031Ra 4
3.5
3
2.5
2
1.5
0.4 0.45 0.5 0.55
WHEREρ ISTHERELATIVEDENSITYOFIRONCOMPACT ATTHEPOINTWITHMAXIMUM+))
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3INCE NO PRECRACK IS PRESENT IN THIS CASE THE DIRECTION OF MAXIMUM SHEAR STRESS IS USED AS THEORIGINALCRACKDIRECTIONINTHECALCULATIONOF +)θ AND+))θ FORTHElRSTCRACKFORMATION4HIS IS ACCEPTABLE BECAUSE THE SAME CONCLUSIONS REGARDING THE CRACK PATH ARE ACHIEVED IN MATERIALS UNDER COMPRESSION BY ASSUMING THATCRACKGROWSALONGTHEPLANEOFMAXIMUM SHEAR STRESS AS BY ASSUMING THAT CRACK FOLLOWS THEDIRECTIONOFMAXIMUM+)))SAKSSON3TAHLE 7ITHOUTPRECRACKINTHISWORKTHEPOINT WITHMAXIMUMSHEARSTRESSISTAKENASTHEPOINT WHERETHECRACKSTARTS
! SINGLE CRACK INITIATED FROM THE BOUNDARY SURFACE IS CONSIDERED IN THIS WORK 3IMULATION OF THE FRACTURE PROCESS SHOW THAT SHEAR CRACK STARTSATTHEENDOFSTEPWHEN-OHR#OULOMB YIELDCRITERIAISUSED4HESHEARSTRESSANDRELATIVE DENSITY DISTRIBUTIONS AT STEP ARE SHOWN IN
&IGURES A AND A RESPECTIVELY &ROM THESE TWOlGURESITCANBESEENTHATCRACKSTARTSINTHE REGIONWITHTHEHIGHESTSHEARSTRESSDISTRIBUTION BUTLOWESTRELATIVEDENSITYDISTRIBUTION3IMILARLY
&IGURESA ANDA SHOWTHATBYUSINGTHE ELLIPTICALCAPYIELDCRITERIASHEARCRACKSTARTSAT A POINT WITH MAXIMUM SHEAR STRESS BUT IN THE REGIONWHERETHERELATIVEDENSITYISMUCHLOWER )NTHISCASEHOWEVERCRACKSTARTSATTHEBEGINNING OFCOMPACTIONBYTHETOPPUNCHMOVEMENTSTEP WHERECOMPACTIONBYTHEBOTTOMPUNCHHAD COMPLETECAUSINGTHELOWERPARTOFCOMPACTED POWDER TO HAVE MUCH HIGHER RELATIVE DENSITY DISTRIBUTIONASSHOWNIN&IGUREA
!SCOMPACTIONPROCEEDSSIMULATIONSOFCRACK PROPAGATIONSHOWTHATDIFFERENTCRACKPATTERNS ARE OBTAINED BY USING DIFFERENT YIELD CRITERIA
#RACKPROPAGATIONDIRECTIONSBYUSINGTHETWO YIELDCRITERIAARELISTEDIN4ABLEINCLUDINGTHE RELATIVE DENSITY AT THE POINT WHERE CRACK lRST STARTSTOPROPAGATE.OFURTHERCRACKPROPAGATION OCCURSAFTERSTEPWHEN-OHR#OULOMBCRITERIA ISUSEDWHILECRACKPROPAGATESONLYONCEWHEN ELLIPTICALCAPCRITERIAISUSED
4HE VALUE OFθ IS CALCULATED WITH RESPECT TO THE EARLIER CRACK DIRECTION USING THE SIGN CONVENTIONASSHOWNIN&IGUREWHILEρISTHE RELATIVEDENSITYVALUEATTHEPOINTWHERECRACK PROPAGATES4HESEVALUESOFθASLISTEDIN4ABLE
SHOW THAT THE CRACK ALWAYS EXTENDED AWAY FROMTHEORIGINALCRACKDIRECTION.EGLECTINGTHE SIGN CONVENTION WHICH INDICATES THE DIRECTION OF STRESSES IT CAN BE SEEN FROM &IGURE A TO
&IGURE B AND &IGURE A TO &IGURE B THAT THECRACKPROPAGATESINTHEDIRECTIONWHERETHE SHEARSTRESSANDRELATIVEDENSITYAREMUCHHIGHER 3INCE RELATIVE DENSITY INCREASES AS COMPACTION PRESSUREINCREASES!IDAH0OQUILLONETAL IT CAN BE DEDUCED THAT THE CRACK GROWS INTHEDIRECTIONOFHIGHERCOMPACTIONPRESSURE 4HISISINLINEWITHTHECONCLUSIONMADEBY!RUN 2OYETAL WHICHARGUETHATCRACKGROWS INTHEDIRECTIONOFHIGHERCONlNINGHYDROSTATIC PRESSUREORCOMPACTIONPRESSUREINTHISCASE "YUSINGTHEELLIPTICALCAPYIELDCRITERIAWHICH PERMITS REPRESENTATION OF DENSIlCATION AS WELL AS HARDENING THE PREDICTED CRACK PROPAGATION DIRECTION IS ONLY≈ O AWAY FROM THE ORIGINAL CRACKPLANE4HISISALMOSTINTHEPLANEOFORIGINAL CRACK DIRECTION AS SHOWN IN &IGURE B AND
&IGUREB )NADDITIONTHECRACKSTARTSATTHE ENDOFSTEPANDPROPAGATESATSTEPWHICHIS MUCHLATERCOMPAREDTOTHEPREVIOUSCASEWHERE CRACKSTARTSATSTEPWHEN-OHR#OULOMBYIELD CRITERIAISUSED
4HENUMERICALANDEXPERIMENTALREPORTSBY PREVIOUSRESEARCHERSONLYPROVIDETHEANALYSISIN TERMSOFCRACKDIRECTIONWHENTHECRACKlRSTSTARTS TO PROPAGATE (ENCE FOR COMPARISON PURPOSES THEVALUEOFRELATIVEDENSITYATWHICHTHECRACK lRSTSTARTSTOPROPAGATEISGIVENIN4ABLEWHEN TWODIFFERENTYIELDCRITERIAAREUSED)TCANBESEEN FROM4ABLETHATTHECRACKPROPAGATIONDIRECTION ISMUCHNEARERTOTHEORIGINALCRACKPLANEWHEN CRACKSTARTSTOPROPAGATEATAPOINTWITHMUCH HIGHER RELATIVE DENSITY WHICH ALSO IMPLIES A MUCHHIGHERCOMPACTIONPRESSURE4HISISINDEED AGREEDVERYWELLWITHTHECONCLUSIONBY)SAKSSON AND3TAHLE WHICHSTATEDTHATTHEDIRECTION OFTHECRACKGROWTHWITHRESPECTTOTHEORIGINAL CRACKPLANEDECREASESWITHINCREASINGPRESSURE $UETOTHEMOVEMENTOFTHEBOTTOMPUNCH IN THE lRST STEPS THE TWO REGIONS WITH THE HIGHESTANDLOWESTRELATIVEDENSITYAREFORMED AROUNDTHESHARPCORNERPROVIDINGAHIGHDENSITY GRADIENTANDCAUSESTHECRACKTOPROPAGATESIN SUCHDIRECTIONASSHOWNIN&IGUREAND&IGURE 4HESHEARSTRESSESWHICHAREINITIALLYHIGHER AT THE BOUNDARY AS SHOWN IN &IGURE A AND
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!DISPLACEMENTBASEDlNITEELEMENTMODELHAS BEENDEVELOPEDTOANALYSETHECRACKINITIATION
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%NGINEERING COMPACTION STEP AND DIFFERENT CRACK PATTERNS
ARE OBTAINED WHEN DIFFERENT YIELD CRITERIA IS USED CRACK IS PREDICTED TO STARTS AT ABOUT THE SAME POINT NEAR THE SHARP INNER CORNER OF THE COMPONENT IN THE TWO DIFFERENT MODELS !S COMPACTION PROCEEDS THE CRACK PROPAGATES IN THE DIRECTION WHERE THE SHEAR STRESS AND THE RELATIVE DENSITY ARE MUCH HIGHER 0ROPAGATION OFTHECRACKTOWARDSTHEREGIONOFMUCHHIGHER RELATIVE DENSITY DISTRIBUTION ALSO IMPLIES THAT
THE CRACK GROWS IN THE DIRECTION OF HIGHER COMPACTION PRESSURE WHICH IS IN LINE WITH THE CONCLUSION MADE BY PREVIOUS RESEARCHERS ON CRACK GROWTH IN MATERIALS UNDER COMPRESSION
"Y USING THE ELLIPTICAL CAP YIELD CRITERIA WHICH PERMITSREPRESENTATIONOFDENSIlCATIONASWELLAS HARDENINGTHECRACKPROPAGATESONLYONCEAND THECRACKPROPAGATIONDIRECTIONISALMOSTINTHE PLANEOFORIGINALCRACKDIRECTION
