Fatigue life and crack growth direction in 7075-T6 aluminium alloy specimens with a circular hole under biaxial loading

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in International Journal of Fatigue, Vol. 125, on August 2019, , available at: https://doi.org/10.1016/j.ijfatigue.2019.03.031 © 2019 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND 1Introduction Therearecurrentlyseveralmethodstopredictthefatiguelimit(orendurancelimit)innotchedcomponentssubjectedtomultiaxialfatigueloadingsuchasthecriticaldistancemethod(CDM)ofTaylorinconjunctionwiththe ModifiedWöhlerCurvemethod(MWCM)[1]andtheextensionoftheshortcrackmodelofNavarroanddelosRios(N-Rmodel)[2,3]tospecimenscontainingholesunderin-phasebiaxialloading[4].Thesemethodsarebasedonthe linearelasticstressgradientalongacertainline(focuspath)aheadofthenotch,wherethisfocuspathisrepresentativeoftherealcrackpathinthesolid.Forbothmethods,thelengthofthefocuspathisusuallybetween1and10 grains[5].Inrelationtotheoriginanddirectionofthepath,forsymmetricalnotchesunderaxialloading,astraightlinethatbeginsatthemaximumprincipalstresspoint(hotspot)andisperpendiculartotheholesurfaceatthehot spot,i.e.,perpendiculartothefirstprincipalstress(maximumprincipalstressdirection),hasbeenused.Forbiaxialloading,thechoiceofthefocuspathismorecomplicated.Tayloranalysedthetorsiontestsofspecimenswithholes andfinally,selectedastraightpaththatbeganatthehotspotbutfollowedthedirectionofmaximumshearstress[6].Inhisbook,SusmelproposedthatfortheCDM + MWCMappliedtomultiaxialloading,astraightlinethatemanates fromthehotspotandisperpendiculartothecomponentsurface,atthehotspotitself,i.e.,perpendiculartothemaximumprincipalstressdirection,isused[7].Meanwhile,fortheN-Rmodel,theoriginanddirectionofthepath proposedin[4]arenotprefixedbutcalculatedasthosethatrequiretheminimumappliedstresstofulfiltheactivationcriterion.Insummary,atpresent,thereareseveralfocuspathsforstudyingnotchfatiguelimitsundermultiaxial fatigue. From theexperimentalviewpoint,it isgenerallyacceptedthatthecrackbegins atthehotspot,continues withacrackinitiation phase (Stage I)nearthemaximum shearstressdirection andsubsequentlyproceeds toa propagationphasenearthemaximumprincipalstressdirection(StageII)[8].The transitionpointbetween thetwophaseshasnotbeenwellestablished,althoughitcertainlyoccurs inoneof thefirstgrains:Susmel andTaylor observedthatforcracksthatgrowfromVnotches,theStage-Itransitionoccurredat ( )[1],whereasMurakami'sobservationsincircularholesshowedthatthecrackemanatedperpendiculartothemaximum principaldirection,andtherewasnoevidenceofStageI[9].Otherresearchershaveanalyzedthecrackpathsinnotchedcomponentssubjectedtofatiguemultiaxialloading.Fatemietal.conductedmultiaxialfatiguetestsonthinwalled7075-T6and2024-T3aluminiumtubularspecimenswithacircularholeandobservedthatthedominantcrackstypicallynucleatedonthemaximumshearplaneandquicklyturnedintomodeIdirection [10,11].Atzorietal. studiednotchedspecimensmadeof carbonsteelsubjectedtocombinedloading andfoundthatthecrackpathswereinfluencedby thestresslevelandloadphaseangle[12].Bertoetal.[13] studiedcircumferentially V-notched Fatiguelifeandcrackgrowthdirectionin7075-T6aluminiumalloyspecimenswithacircularholeunderbiaxialloading V.Chaves⁎ chavesr[email protected]s G.Beretta J.A.Balbín A.Navarro DepartamentodeIngenieríaMecánicayFabricación,EscuelaSuperiordeIngeniería,UniversidaddeSevilla,CaminodelosDescubrimientoss/n,41092Sevilla,Spain ⁎Correspondingauthor. Abstract Thisworkshowstheresultsofasetoftestsin7075-T6aluminiumalloyspecimens.Thetestswereunderloadcontrolat .Fromthetests,thecorrespondingS-Ncurveswerebuilt,andtheendurancelimitswere calculatedfor1millioncycles.Theratiobetweenthepuretorsionandtensionendurancelimitswas0.58,i.e.,thebehaviourofthismaterialinfatigueisofthevonMisestype.Specimenswithcircularholesofvarious diametersundertension,torsionandin-phasebiaxialloadingweretested.Thedirectionsofthecracksthatgrewfromtheholeswerestudiedwithanopticalmicroscope,ascanningelectronmicroscope(SEM)andanoncontact3Dopticalprofiler.Inparticular,thepointswherethecracksbeganandthedirectioninwhichthecrackspropagatedalongthefirst150 μmwereanalysed.Ingeneral,theinitiationpointwasclosetothemaximum principalstresspoint,andthecrackdirectionwasclosetothemaximumprincipalstressdirection.Theendurancelimitpredictionswereclosetotheexperimentalresults. Keywords:Biaxialtest;Highcyclefatigue;Notch;Endurancelimit;Crackdirection  R =-1     specimensmadeofhardenedandtemperedsteelsubjectedtocombinedtensionandtorsionloadingandexaminedthepathsofnon-propagatingcracksfromsomerun-outspecimens.Undertensileloadingthecrackspropagatedmainly fromthenotchtipalongthenotchbisector.Tanakaetal.[14]testedthin-walledtubularspecimensofmedium-carbonsteelwithahole,undercombinationsofcyclictorsionandaxialloadings,andthecrackswerenucleatedatthepoint ofmaximumcircumferentialstressontheperipheryoftheholeandpropagatedalmoststraightforallcasesexamined.non-propagatingcracks.Matsunagaetal.[15]analysedtheformationofnon-propagatingcracksinabearingsteel withseveraladjacentholessubjectedtofullyreversedtorsioncombinedwithstaticaxialcompressivestressandobservedthatcrackswerenotasstraightasalinebutlocallykinked.Carpinterietal.[16]extendedtheirmultiaxial criterionusingthecriticaldistancemethodofTaylor[6]toestimatethefatiguelimitofnotchedcomponents.SomeexperimentaltestsbyEndo[17]wereconsideredforcomparison.Theexperimentsconcernedroundbarsmadeof annealedsteelwithdrilledsurfaceholessubjectedtocombinedbendingandtorsion.Thechosenfocuspathtomakethepredictionswiththemodelformedanangleof41°withthenormaltotheholeatthehotspot,i.e.,nearthe maximumshearstressdirection,andthefatiguelimitspredictionswereingoodagreementwiththeexperimentalvalues. Theuseofastraightlineinthefatiguemodelsisobviouslyasimplificationofreality.However,whenoneofthetwophasesoccupiesmostofthefocuspathinthemodels,thesimplificationofusingasinglestraightlineis reasonable.IfStageIoccupiesapproximately5–10grains,thentheappropriatesimplificationforthemodelscanbeastraightpathinthemaximumshearstressdirection.However,ifStageIisverysmallorpracticallyimperceptible, thenitwouldbeconvenienttouseastraightpathinthemaximumprincipalstressdirection. Toanalysetherealcrackpathinnotchedspecimens,particularlyinthefirstgrains,andconsequentlyproposeanappropriatepathforthemodels,anexperimentalcampaignhasbeenestablished,whichbeganwithastudyon AISI304Lstainlesssteelunderproportionalbiaxialloadinginthehigh-cycle-fatigueregime[18,5].Thepresentpaperreportsanexperimentalstudyon7075-T6aluminiumalloy.Thegeometryisathin-walledtubewithapass-through hole,andthetestswereaxial,torsionalandin-phaseaxial-torsional. 2Materialsandexperimentalprocedure The material is commercial 7075-T6 aluminium alloy, a high-strength heat-treated alloy whose typical application is in aircraft structures or light-weight structures that involve high stresses. This material has a good combinationoffatigueresistanceandlowenvironmentalimpactfortransportapplications,butthiscombinationisnotgoodforapplicationsthatdonotrequireenergyduringitsuse[19].Itschemicalcomposition(inwt.%)isas follows:6.05Zn,2.83 Mg,1.21Cu,0.14Fe,0.26Cr,0.04Si,0.028Ti,and0.023Mn.Theaveragegrainsizeis15 μm,ascalculatedbytheASTMstandardE112-96[20](seethemicrostructureinFig.1).Noheattreatmentisapplied afterthemachiningofthespecimens.Themonotonicmechanicalpropertiesaredeterminedfrom3tensiletests,asfollows:tensilestrength MPa;yieldstrength MPa. Fatiguetestsunderfullyreversedloading wereconductedonsmoothcylindricalspecimenswithasolidcircularcross-section.Thecentraldiameter was7 mmfortheaxialtests(Fig.2a)and14 mmforthetorsional tests(Fig.2b).Theaxialtestswereperformedinaresonancetestingmachineatafrequencyof100–120 Hz,whereasthetorsionaltestswereperformedinaservo-hydraulicaxial-torsionloadframeat6–8 Hz.Thespecimenswere carefullypolishedtoanaverageroughness ofapproximately0.13 μm.  σUTS =657  σYS =595 Fig.1Microstructureofthe7075-T6aluminiumalloy.    ds   Ra  Thegeometryofthenotchedspecimenwasathin-walledtubeof1.5 mmthicknesswithapass-throughholeinthecentralsection.Therewere3differentholediameters :1,2and3 mm.Fig.2cshowsthegeometryofthe specimens,particularlytheonewiththehole mm.Theaxialtestswereperformedintheresonancetestingmachine,andthetorsionalandbiaxialtestswereperformedintheservo-hydraulicmachine. 3Experimentalresults Theresultswereusedtoconstructthe curvesinaccordancewithASTMstandardE739[21], assuminga linearrelationshipbetween thevariables and . Fig. 3 shows the logarithmic diagrams of the experimental curvesfortheplainspecimens.ThefatiguetestconditionsandresultsisprovidedintabulatedforminAppendixA.Thetestsresultsinthetablearenotchronologicallyorderedbutindescendingorderofthestress magnitude.Theendurancelimitat cycles,expressedintermsofstressamplitude,was MPafortheaxialtestsand MPaforthetorsionaltests.Thesetwostresseswerecalculatedbytheequationsofelasticity: and ,where istheappliedforceamplitude, istheappliedtorqueamplitudeand isthecentraldiameter.Thismaterialhas ,accordingtothevonMisescriterion. Fig.2Geometryofthespecimens(inmm):(a)plainspecimentestedinaxialfatigue,(b)plainspecimentestedintorsionalfatigue,(c)notchedspecimenwiththepassingthroughhole(holediameter mm). d =1  d   d =1  S - N       S - N  106  σFL =258  τFL =149     F   T   ds    Sixsetsoftestswereperformedwiththenotchedspecimens:push-pulltestsforholediameters ,2and3 mm;torsiontestsfor and2 mm ;in-phaseaxial-torsiontestsfor mm ofthetype ,where isthenormalstressamplitudeand isthemaximumshearstressamplitude,bothofwhichwereatthecentraltransversecross-section.Thesetwostresseswerecalculatedconsideringthegrossarea,i.e.,neglecting the reduction of the cross-section by the presence of the hole. This simplification is justified on the basis that the hole transverse cross-section is much smaller than the specimen transverse cross-section. Figs. 4–6 show the experimental curvesforthesixsetsoftests.TheresultsarealsotabulatedinAppendixA. Fig.3S-Ncurvesfortheplainspecimens(R = −1):push-pullandtorsion.  d =1  d =1    d =1   σ = τ   σ   τ   S - N  Fig.4S-Ncurvesforthenotchedspecimens, mm :push-pull,torsionandbiaxial . d =1    σ = τ  Fig.5S-Ncurvesforthenotchedspecimens, mm :push-pullandtorsion. d =2   4Crackpaths As discussed in the Introduction, it is important to determine the experimental point of crack initiation and the crack path, particularly during the first period of crack growth, because some fatigue models base their evaluationsonthestresses along alinethatshould berepresentativeofthecrackpath.Thissection addressestheexperimentally determinedcrack initiationpointatthe hole surfaceandcrackdirection.Thecrackdirection is consideredastraightline,sincethesemodelscannothandleacrackthatfollowsazig-zagpath.TheX-axisisinthecentraltransversesectionofthespecimen,theY-axisisinparallelwiththespecimen,andtheZ-axiscoincideswith theaxisofthehole,asshowninFig.7.Theorigin of thecoordinate system islocated onthe specimen surface.Thevariables and are measured at the specimen surface ( ) from photographs of the hole and its surroundingtakenwithalightmicroscope.Theangle hasbeencalculatedbyjoiningcentre oftheholeandthepointofthecrackattheholesurfacewithastraightlineandmeasuringitsanglewithrespecttotheX-axis.Similarly, angle hasbeencalculatedfromthestraightlinebuiltwiththecrackinitiationpointandthecrackpointat150 μmaheadoftheholesurface.Thislengthof150 μmisapproximately10averagegrainsforthismaterial,whichis representativeofthelengthofthefocuspathinsomemodelstopredictthefatiguelimit(orendurancelimit)innotchedcomponents. Fig.6S-Ncurveforthenotchedspecimens, mm :push-pull. d =3    O   θ   θ 1  Z  = 0  θ   O   θ 1 TheexperimentalcrackinitiationpointsandcrackdirectionsisprovidedintabulatedforminAppendixB.Thevariablesforthecracksononesideoftheholearetermed and andthoseontheothersidearetermed and ,whicharepresentedasabsolutevalues.Onlythefailuresthatoccurredinthehigh-cyclefatigueregime(foranumberofcyclestofailurehigherthan cycles)wereconsidered.Theaveragevalues and andrelative standarddeviations(numbersinbrackets)oftheexperimentalvaluesof and fortheaxial,biaxialandtorsionaltestsanddifferentdiametersarealsopresentedinthetableofAppendixB.Fortheanalyseddiameters and 3 mm,theresultsshownonoticeabledifference.However,theaveragevalues​​ and clearlyvaryforthethreetypesofloading:theaxialtestshadthelowestvalues,followedbythebiaxialtests,andthetorsionaltestshadthe largestvalues.Alinearelastic2Danalysisindicatesthatthemaximumprincipalstresspoint(hotspot)islocatedat foraxialloading, forbiaxialloading and fortorsionalloading.Theseangleswillbecalled .Theaverageexperimentalresults arenotsignificantlydifferentfromthesevalues,particularlyforthetorsionaltests.Themaximumprincipalstressdirectionfromthehotspotisat foraxial, forbiaxial and fortorsion.Theseangleswillbecalled .Again,theaverageexperimentalvalues ​​arenotsignificantlydifferentfromthesevalues,particularlythoseoftorsion.Forthe46analysedtests,the averagedifferencebetweentheexperimentalangles andthecorrespondingangles ,is ,andtheaveragedifferencebetweentheexperimentalangles andthecorrespondingangle is .Thus,from thetests,wededucethatthecrackinitiationpointonthespecimensurfaceisnearthehotspot,andthecrackdirectiononthespecimensurfaceinthefirst150 μmisrelativelynearthemaximumprincipalstressdirection. Fig.8showsrepresentativepicturesofthecracksatthespecimensurface( )forapush-pulltest,abiaxialtestandatorsionaltest.Twoopposingcracksformedinthemostcases.Asshowninthefigure,thecrack Fig.7Sketchofthenotchedspecimenwiththeaxesandanglesforthestudyofthecrackinitiationpointandthecrackdirection:(a)thewholespecimenandthecrosssection,(b)detailofthehole.  θa     θb   105      θ   θ 1  d =1,2      θ =0∘  θ =31.7∘  σ = τ   θ =45∘      θ 1=0∘  θ 1=31.7∘ σ = τ   θ 1=45∘      θ    5.6∘  θ 1   10.9∘  Z =0 initiationpointandcrackdirectionisclearlydifferentamongthethreeloadcases.Thecrackinitiationpointsarenearthepointsofmaximumprincipalstress,i.e., , and fortheaxial,biaxialandtorsioncases, respectively.Thecrackdirectionsalongthefirst150 μmwereclosetothemaximumprincipalstressdirection,i.e., , and fortheaxial,biaxialandtorsioncases,respectively. Theanglesweremeasuredonthespecimensurface( ),butthecrackmightactuallystartataninteriorpointonthesurfaceofthehole( ),whichhasadepthof1500 μm.Thepointofmaximumprincipalstressofthe entirespecimenwascalculatedwithafiniteelementanalysis(commercialsoftwareANSYSv.15.0[22])forthethreetypesofloading.SOLID187elementsanddifferentmeshsizeswereusedinalinearelasticanalysis.Themodels werere-meshedtoensurethatthedifferencebetweenthestressesofamodelandthepreviousonewerepracticallynegligible.Thefinalmeshhadanelementsizeofapproximately2 mm,beingrefinedattheholearea,uptoan elementsizeof0.025 mm.Fig.9showsthemodelgeometryandmeshandthedetailofthemesharoundtheholeforthe mmcase.Asfortheboundaryconditions,fortheaxialcasethespecimenwasclampedatthebottom surfaceandapressurewasappliedontheuppersurface.Forthetorsionalcase,atorquewasappliedontheuppersurface.  θ =0∘  θ =31.7∘  θ =45∘  θ 1=0∘  θ 1=31.7∘  θ 1=45∘ Fig.8Crackpathsatthespecimensurface( ):(a)Push-pulltest, mm, MPa, cycles;(b)In-phasebiaxialtest, mm, MPa, cycles;(c)Puretorsiontest, mm, MPa, cycles.  Z =0  d =1  σ =68    d =1  σ = τ =60    d =1  τ =80   Z =0  Z <0  d =1 Fig.10shows,forthemodelwithaholeof mm,themaximumprincipalstress forpureaxial,puretorsionalandbiaxial loadings,alongthedepthofthehole(pathX = 500 μm).Theappliedstressistheunitinall cases.Inthegraph,forthesakeofclarity,thestress isshowndimensionless,dividingbythe ofaholeinaninfinitemedium. Accordingtothisanalysis,thispointisnotlocatedatthespecimensurface( )butat μm, μmand μmforthecaseofaxialloading,biaxialloadingandtorsionalloading,i.e.,attheinternalsurfaceof thehole.Apossibleexplanationofthistypeofstressdistributionina3Dholeisshownin[11]. Tostudythecrackintheseinternalplanes,weremovedthematerialinseveraltestedspecimensuntilreachingtheseplanes.Fig.11showsthecrackpathsat μm, μmand μmfortheaxialtest,biaxialtest andtorsionaltest,respectively.ThespecimenswereidenticaltothosepreviouslystudiedandshowninFig.8.Asshowninthesiximages,theinitiationpointsandcrackdirectionsfollowasimilartrendintheoutersurface( )and internalplanes( )forthethreetypesofloading.Thus,angles and obtainedonthespecimensurfacearegenerallyrepresentativeofthecrackanglesatinternalplane ,wherethepointofmaximumprincipalstressis located.Acarefullookattheimagesofthespecimensubjectedtoaxialloading(Figs.8aand11a)showsthattheinitiationpointsat and (leftcrack)areslightlydifferent.Themostlikelyjustificationforthisdifferenceis thatthesetwoinitiationpointsbelongtotwodifferentcracks,whichformedatdifferentheightsandprobablysubsequentlyjoinedeachother.Thefiniteelementanalysisofthespecimenwiththe1-mmholeshowsthatfortheaxial loading,themaximumprincipalstressalongthelines μm(attheholesurface)didnotsignificantlychange.Infact,foralengthof810 μmalongthesetwolines,themaximumprincipalstressdifferedlessthan with respecttoitsmaximumat μm.Hence,thecrackscouldhaveinitiatedfromalmostanypointalongtheplanes μm,whichjustifiestheexistenceofseveralinitiationpointsatdifferentvaluesof .Toanalysethese aspectsinmoredetailandprecision,itisnecessarytoanalysethefracturesurface,asshowninthenextsection. Fig.9Specimenmodelandmeshusedinfiniteelementanalysis.Detailofthemesharoundthehole.  d =1  S 1  σ = τ   S 1  kt  Fig.10Elasticstressdistributionalongtheholedepth(path μm),forpureaxial,torsionalandbiaxial loading. mm. X =500  σ = τ  d =1  Z =0  Z =-750 -450 -350  Z =-750 -450 -350  Z =0  Z <0  θ   θ 1  Z   Z =0  Z =-750  X =±500 1%  Z =-750  X =±500  X  Forthecrackdirectionsattheplaneswherethemaximumprincipalstresspointsarelocated,Path2inFigs.15–17showtheheightsofthepointsalongthefirst150 μmintheplanes μm, μmand μm (forthepush-pulltest,biaxialtestandtorsionaltest,respectively).Thedifferenceinheightinthesepathsalongthe150 μmis ,2and41 μm,respectively.Theanglesofthesepathswithrespecttothemaximumprincipalstress directionscanbecalculatedusingtheequation andhavethevaluesof , and ,givinganaveragevalueof .Then,thecrackdirectionsintheinternalplanesthatincludethemaximumstresspointin thefirst150µmarerelativelynearthemaximumprincipalstressdirections. 6Fatiguestrengthpredictions ThecomparisonbetweentheexperimentalresultsandthepredictionsoftheN-RmodelandtheCDM + MWCMisshownbelow.Thisanalysisisdonefortheendurancelimitat1millioncyclesofnotchedspecimens,definedas foraxialorbiaxialloadingandas fortorsionalloading.Briefly,fortheN-Rmodel,thebiaxialversionhasbeenused,withthefollowingparameters:endurancelimitsfortheplainspecimens( MPa, MPa), grainsize mm, mm(approximatedbyusingthemicrostructuralequationfrom[23]),KitagawadiagramapproximatedwiththeequationproposedbyVallelanoetal.,using [24].TheN-Rbiaxialcriterionisas follows[4]: y areorientationfactorsthatprojectthenormalandshearstress and inthebarrierareaintotheslipplaneandtheslipdirectionofthedislocationsources,and isthecriticalstressnecessarytoactivatethe dislocationsources.FortheCDM + MWCM,thePointmethod(PM)hasbeenused,with mm,andtwofocuspathshavebeenstudied:the path,whichisapaththatstartsatthehotspotandfollowthedirectionof maximumshearstress,proposedbyTaylor[6],andthe path,apaththatstartsatthehotspotandfollowthedirectionofmaximumprincipalstress,proposedbySusmel[7].TheCDM + MWCMcriterionis: Fig.17Heightsofthepointsofthefracturesurfacealongtwoplanesforatorsionaltest:Path1, μm;Path2, μm. X ≈500  Z =-350  Z =-750  Z =-450  Z =-350 Δ h =14   1.6∘ 0.8∘ 15.8∘ 6.1∘     σFL =258  τFL =149  D =0.015  a 0=0.072  f =2.5 (1)          τc   L = a 0=0.072  θ =0∘  θ =45∘ where is the shear stress amplitude on the critical plane and is the maximum tensile stress normal to this plane. Table 1 show the experimental results and the predictions provided by the different models. The endurancelimitpredictionerrorhasbeencalculatedinthefollowingway: Table1Predictionswithseveralmultiaxialmodelsfornotches. Loading Holediameter(mm) Endurancelimit Experimental N-Rmodel PM + MWCM( path) PM + MWCM( path) (MPa) (MPa) Error( ) (MPa) Error( ) (MPa) Error( ) Axial 1 95 92 −2.7 103 8.4 98 3.4 2 94 92 −2.1 94 0.4 92 −2.1 3 85 92 6.4 92 7.8 90 5.9 Biaxial 1 65 68 3.9 52 −19.6 55 −16.6 Torsion 1 75 69 −6.1 80 7.2 76 1.2 2 67 69 2.5 72 7.9 70 4.5 Average: 4.0 8.6 5.5 Allthemodelsgivereasonablepredictions,especiallytheN-Rmodelandthe path.Errorsarebelow ,exceptintwocases,whicharebelow .Inrelationtotheoriginofthepath,theCDM + MWCMhasit prefixedinthehotspotforanytypeofloading,whichisclosetotheaverageexperimentalvalues,whilefortheN-Rmodelithasbeencalculated,obtainingvalues​​of , ,and ,foraxial,biaxialandtorsional loading,respectively,beingalsoclosetotheexperimentalones.Asforthedirectionofthepath,thefirstprincipalstressdirectionsusedinthe pathareclosetothecrackdirectionsobtainedexperimentallyinthefirst150µm. However,thedirectionsofthe path,thoseofmaximumtangentialstresses,arefarfromtheexperimentalones.ThedirectionscalculatedwiththeN-Rmodelare , ,and ,foraxial,biaxialandtorsional loading,respectively,whicharealsofarfromtheexperimentalones. 7Conclusions Highcyclefatiguetestswereconductedfor7075-T6aluminiumalloyspecimenswithcircularholes.Thetestswereaxial,torsionalandin-phaseaxial-torsional,allofwhichwereunderloadcontrolwith .Thecrackpaths ofthebrokenspecimenswereanalysedwithalightmicroscope,ascanningelectronmicroscope(SEM)andanon-contact3Dopticalprofilerwithspecialattentiontotheinitiationpointandthedirectionofthecrackalongthefirst 150 μminfrontofthehole.Thislengthwasapproximately10grainsforthismaterial,i.e.,oftheorderofmagnitudeofthelengthsforthepredictionsofmethodssuchasthecriticaldistancemethod + ModifiedWöhlerCurvemethodor theN-Rmodel.Theexperimentalresultsindicatethatforthethreetypesofloading,thecrackinitiationpointwasnearthepointofmaximumprincipalstressattheholesurface,andthecrackdirectionalongthefirst150 μmwasnear themaximumprincipalstressdirection.Then,atleastforthismaterial,thelinethatshouldbeusedtomakethepredictionswiththemodelsbeginsatthepointofmaximumprincipalstressattheholesurfaceandfollowsthemaximum principalstressdirection.Theanalysedmultiaxialmodelsgavegoodendurancelimitpredictions,themaximumerrorbeinglessthan ,withmostpredictionswithin oftheexperimentaldata. Acknowledgements TheauthorswouldliketothanktheSpanishMinistryofEducationforitsfinancialsupportthroughgrantDPI2017-84788-P. AppendixA (2)  τa   σn , max  (3) 0∘45∘     %   %   %   θ =45∘ 10% 20%  θ =0∘  θ =27∘  θ =44∘  θ =0∘  θ =45∘  θ 1=34∘  θ 1=76∘  θ 1=78∘  R =-1 20% 10% Summaryoffatiguetestconditionsandresultsfor7075-T6aluminiumalloy : (a)Cylindricalsmoothspecimens: (b)Thin-walledtubewithapassingthroughholeofdiameter (mm):   Test (MPa) (MPa) N(cycles) Remarks Axial 310 0 84700 Failed 300 0 84800 Failed 290 0 157500 Failed 290 0 181300 Failed 280 0 178320 Failed 270 0 238400 Failed 270 0 253300 Failed 260 0 403500 Failed 260 0 1363000 Failed 250 0 1387700 Failed 250 0 2263800 Failed 240 0 4279500 Failed Torsional 0 220 10100 Failed 0 200 32900 Failed 0 180 81800 Failed 0 170 285400 Failed 0 160 437000 Failed 0 155 1185600 Failed 0 150 888600 Failed 0 140 1055900 Failed  σ   τ   d  Test (mm) (MPa) (MPa) N(cycles) Remarks Axial 1 135 0 46800 Failed 127 0 58000 Failed 120 0 361700 Failed 117 0 101700 Failed 107 0 145200 Failed 98 0 323400 Failed 95 0 285900 Failed  d   σ   τ  88 0 2417400 Failed 80 0 2544700 Failed 78 0 17071900 Failed 68 0 29318000 Failed 2 140 0 69900 Failed 120 0 107400 Failed 115 0 114500 Failed 110 0 139900 Failed 110 0 341800 Failed 105 0 113600 Failed 100 0 875500 Failed 100 0 2135100 Failed 100 0 3309600 Failed 96 0 145100 Failed 90 0 21398000 Failed 86 0 256400 Failed 86 0 313600 Failed 86 0 8463100 Failed 80 0 1556900 Failed 76 0 499500 Failed 76 0 510100 Failed 3 120 0 79800 Failed 105 0 188300 Failed 100 0 45300 Failed 95 0 243700 Failed 90 0 140900 Failed 90 0 378100 Failed 80 0 417300 Failed 80 0 839000 Failed 80 0 5220100 Failed 75 0 5480500 Failed 70 0 17237000 Failed Biaxial : 1 90 90 18400 Failed  σ = τ  AppendixB Experimentalcrackinitiationpoints( )andcrackdirections( )for7075-T6aluminiumalloy ,inabsolutevalue.Thin-walledtubewithapassingthroughholeofdiameter (mm): 90 90 26000 Failed 80 80 67500 Failed 80 80 130400 Failed 70 70 77300 Failed 70 70 215400 Failed 70 70 520900 Failed 65 65 2154300 Failed 60 60 1430900 Failed 60 60 4113600 Failed Torsion 1 0 115 23500 Failed 0 110 35400 Failed 0 100 64000 Failed 0 95 102700 Failed 0 90 313900 Failed 0 85 427400 Failed 0 80 212200 Failed 0 80 290000 Failed 0 75 1533300 Failed 2 0 100 30100 Failed 0 90 49200 Failed 0 90 53800 Failed 0 80 30800 Failed 0 80 68200 Failed 0 80 788700 Failed 0 70 90500 Failed 0 70 149900 Failed 0 60 115600 Failed  θ   θ 1    d  (mm) (MPa) N (cycles) (°) (°) Average, (°) (°) (°) Average, (°) (a)Axial  d  σ , τ  θa  θb     1 120 361700 1.1 11.7 2.1 22.1 117 101700 0.0 1.6 17.0 7.4 107 145200 4.4 12.8 19.9 4.8 98 323400 5.9 1.1 39.9 0.0 88 2417400 4.9 7.6 6.9(4.8) 14.3 14.0 14.2(11.8) 80 2544700 10.0 16.9 39.0 12.0 78 17071900 11.1 7.1 16.8 2.3 68 29318000 9.5 5.1 7.3 8.8 2 120 107400 10.6 5.3 17.8 10.5 115 114500 0.2 10.0 13.1 13.1 110 139900 5.0 1.5 7.9 9.1 110 341800 0.7 9.7 6.5 9.7 105 113600 2.3 15.0 11.1 39.1 100 875500 2.1 3.5 3.2 2.6 100 2135100 4.1 5.8 9.8 2.0 100 3309600 15.0 7.9 24.2 5.3 96 145100 8.7 4.4 32.6 3.8 90 21398000 4.8 9.6 2.6 5.9 86 256400 2.9 16.7 20.9 7.2 86 313600 6.4 1.4 6.4(4.6) 24.8 2.0 12.3(9.6) 86 8463100 3.7 11.7 3.9 3.3 80 1556900 5.2 3.7 11.7 26.5 76 499500 1.9 15.3 7.2 23.6 76 510100 3.7 7.0 18.4 15.2 3 105 188300 4.7 9.7 9.8 4.6 95 243700 6.1 6.5 3.6 8.9 90 140900 8.4 12.9 26.6 20.4 90 378100 14.4 0.9 14.5 10.8 80 417300 13.0 19.1 8.3(5.5) 30.5 11.5 15.3(11.7) 80 839000 3.2 15.2 12.5 8.9 80 5220100 2.2 5.6 17.4 11.5 75 5480500 15.3 7.3 46.3 5.2 70 17237000 4.4 0.6 31.3 1.8 (c)Biaxial References [1]L.SusmelandD.Taylor,Twomethodsforpredictingthemultiaxialfatiguelimitsofsharpnotches, FatigueFractEngngMaterStruct 26,2003,821–833. [2]A.NavarroandE.R.delosRios,Shortandlongfatiguecrackgrowth:aunifiedmodel, PhilosMagA 57,1988,12–36. [3]A.NavarroandE.R.delosRios,Analternativemodeloftheblockingofdislocationsatgrainboundaries, PhilosMagA 57,1988,37–42. [4]V.Chaves,A.Navarro,G.BerettaandC.Madrigal,Microstructuralmodelforpredictingfatiguestrengthinthepresenceofholesunderproportionalbiaxialloading, TheorApplFractMec 73,2014,27–38. [5]V.Chaves,G.BerettaandA.Navarro,Biaxialfatiguelimitsandcrackdirectionsforstainlesssteelspecimenswithcircularholes, EngFractMech 174,2017,139–154. [6]D.Taylor,Thetheoryofcriticaldistances:anewperspectiveinfracturemechanics,2007,Elsevier. [7]L.Susmel,Multiaxialnotchfatige:fromnominaltolocalstress/strainquantities, CRC 2009. [8]ForsythPJE.Atwostageprocessoffatiguecrackgrowth.In:ProceedingsoftheCrackPropagationSymposium,CranfieldCollegeofAeronautics,Cranfield,UK,vol.1,p.76–94,1962. [9]Y.Murakami,Metalfatigue:effectofsmalldefectsandnonmetallicinclusions,2002,Elsevier. [10]A.Fatemi,N.Gates,D.SocieandN.Phan,Fatiguecrackgrowthbehaviourontubularaluminiumspecimenswithacircularholeunderaxialandtorsionloadings, EngFractMech 123,2014,137–147. [11]N.GatesandA.Fatemi,Notchedfatiguebehaviourandstressanalysisundermultiaxialstatesofstress, IntJFatigue 67,2014,2–14. [12]B.Atzori,F.Berto,P.LazzarinandM.Quaresimin,Multi-axialfatiguebehaviourofaseverelynotchedcarbonsteel, IntJFatigue 28,2006,485–493. [13]F.Berto,P.LazzarinandJ.R.Yates,MultiaxialfatigueofV-notchedsteelspecimens:anon-conventionalapplicationofthelocalenergymethod, FatigueFractEngngMaterStruct 34,2011,921–943. [14]K.Tanaka,K.MoritaandY.Akinawa,Fatiguethresholdsofholedcomponentsunderin-phaseandout-of-phasetorsionalandaxialloadings, FatigueFractEngngMaterStruct 31,2008,1079–1090. [15]H.Matsunaga,N.Shomura,S.MuramotoandM.Endo,Shearmodethresholdforasmallfatiguecrackinabearingsteel, FatigueFractEngngMaterStruct 34,2010,72–82. 1 80 130400 31.8 22.8 30.3 39.3 70 215400 38.5 18.6 28.8 37.9 70 520900 39.5 34.1 29.2(7.6) 42.8 37.4 36.6(5.4) 65 2154300 29.8 17.3 31.4 35.3 60 4113600 28.2 31.5 37.4 45.8 (b)Torsion 1 95 102700 52.9 60.0 39.1 49.9 90 313900 41.2 25.9 50.5 42.4 85 427400 34.4 37.9 42.4(10.4) 57.2 35.1 47.4(7.2) 80 212200 33.3 40.2 51.3 55.9 80 290000 46.6 51.5 49.2 43.0 2 80 788700 42.6 37.3 46.4 36.6 70 149900 44.8 53.1 47.7(7.8) 45.5 42.7 50.3(12.3) 60 115600 49.2 59.2 63.7 67.2 QueriesandAnswers Query:YourarticleisregisteredasbelongingtotheSpecialIssue/Collectionentitled“FATIGUECRACKPATHS2018”.IfthisisNOTcorrectandyourarticleisaregularitemorbelongstoadifferent SpecialIssuepleasecontact[email protected]immediatelypriortoreturningyourcorrections. Answer:Yes Query:Theauthornameshavebeentaggedasgivennamesandsurnames(surnamesarehighlightedintealcolor).Pleaseconfirmiftheyhavebeenidentifiedcorrectly. Answer:Yes [16]A.Carpinteri,A.Spagnoli,S.VantadoriandD.Viappiani,Amultiaxialcriterionfornotchhigh-cyclefatigueusingacritical-pointmethod, EngFractMech 75,2008,1864–1874. [17]M.Endo,Effectsofsmalldefectsonthefatiguestrengthsofsteelandductileironundercombinedaxialtorsionalloading,In:K.S.Ravichandran,R.O.RitchieandY.Murakami,(Eds.), Smallfatiguecracks:mechanics. Mechanismsandapplications, 1999,Elsevier,375–387. [18]V.Chaves,A.NavarroandC.Madrigal,StageIcrackdirectionsunderin-phaseaxial-torsionfatigueloadingforAISI304Lstainlesssteel, IntJFatigue 80,2015,10–21. [19]V.Chaves,Ecologicalcriteriafortheselectionofmaterialsinfatigue, FatigueFractEngngMaterStruct 37,2014,1034–1042. [20]Standardtestmethodsfordeterminingaveragegrainsize.ASTMStandardE112-96:2004. [21]Standardpracticeforstatisticalanalysisoflinearorlinearizedstress-lifeandstrain-Lifefatiguedata.ASTMStandardE739-91:2004. [22]ANSYSMechanicalAPDL,Release15.0.7. [23]A.Navarro,C.Vallellano,E.R.delosRiosandX.J.Xin,Notchsensitivityandsizeeffectsbyashortcrackpropagationmodel,EngineeringAgainstFatigue,ProcIntConf,1997,AABalkemaPublishers;SheffieldUK. Rotterdam. [24]C.Vallellano,A.NavarroandJ.Dominguez,Fatiguecrackgrowththresholdconditionsatnotches.Part1:theory, FatigueFractEngngMaterStruct 23,2000,113–121. Highlights •7075-T6aluminiumalloywassubjectedtoin-phasebiaxialfatigue. •Thegeometrywasatubewithapassingthroughhole. •TheS-Ncurveswereconstructedandthecracksurfacewasanalyzed. •Thecrackbeganclosetothepointofmaximumprincipalstress. •Thecrackfollowedthemaximumprincipalstressdirection.