OpenCeramics6(2021)100135
ContentslistsavailableatScienceDirect
Open Ceramics
journalhomepage:www.editorialmanager.com/oceram
Kinetics-based constitutive model for self-healing ceramics and its
fi
application to nite element analysis of Alumina/SiC composites
Shingo Ozakia,*, Joji Yamamotoa, Naoki Kandaa, Toshio Osadab
aDivisionofSystemResearch,FacultyofEngineering,YokohamaNationalUniversity,Tokiwadai79-5,Hodogaya-ku,Yokohama,240-8501,Japan
bSuperalloyandHigh-TemperatureMaterialsGroup,ResearchCenterforStructuralMaterials,NationalInstituteforMaterialsScience,Sengen1-2-1Tsukuba,Ibaraki,
305-0047,Japan
A R T I C L E I N F O A B S T R A C T
Keywords: Self-healing ceramics are recognized aspromising next-generation materials owingto theirlight weight, reli-
Self-healing abilityagainstbrittlefracture,andhighcapacitytowithstandextremetemperatures.Thus,abetterunderstanding
Damage ofthesematerialsisnecessarytofacilitatetheiruseascomponents.Inthisstudy,wefirstproposedanoxidation
Finiteelementmethod
kinetics-basedconstitutivemodeltoanalyzebothdamageandhealingprocessesinself-healingceramicswithin
Oxidation theframeworkofthefiniteelementmethod.Inparticular,evolutionlawsfordamagerecoveryduetoself-healing
Kinetics
wereintroduced.Subsequently,weperformedthree-pointbendinganalysesmimickingactualexperimentsfor
considering the self-healing effect under certain temperature and oxygen partial pressure conditions. Our
analyticalresultsconfirmthattheproposedmethodologycanreasonablyreproducebothtime-andenvironment-
dependencies of strength recovery in self-healing ceramics. In this regard, our method can be utilized for
exploringtheself-healingbehaviorlinkedwiththemicrostructuredistributionandfractureproperties,essential
forthematerialdesign.
1. Introduction multipleelementaryprocessessimilartotheinflammation,repair,and
remodelingstagesofbonehealing[25].Further,ithasalsobeenclarified
Recently,attemptshavebeenmadetoimitatetheself-healingfunc- thatthealumina/SiCcompositematerialisanexcellentmaterialsystem
tionofthelivingbodyinartificialindustrialmaterialstoimprovetheir that can simultaneously achieve crack filling and high strength of the
characteristics.Inparticular,introducingthisfunctioninceramics,which healingpart.
arebrittlematerials,is extremelyuseful inensuringhigh reliability of Meanwhile,torealizethepracticalapplicationandwideusageofself-
ceramicmembers[1–19]. healingceramics,backcasting-typematerialdesignisused,inwhichthe
Self-healinginceramicswasintroducedbyLangeetal.[20]inthe idealperformanceofeachproductisdefinedandthematerialisdesigned
1970s as “crack healing by heat treatment.” Further, it has attracted basedontherequiredperformance.Thebackcasting-typeapproachhas
attentionasamethodforimprovingbrittlefractureofceramics.Next, beenmadepossiblebytheexistenceofnumericalanalysismethodson
Andoandcolleaguesproposedself-healingceramicsthatcouldrecover variousscales,theso-calledvirtualtests,inadditiontothesophistication
strength autonomously and completely [21,22]. Subsequently, Osada of experimental observation techniques and the development of data-
et al. [23] demonstrated the repeated recovery of the strength and bases[26–30].Inavirtualtest,thenumericalanalysismethodforeach
fracturetoughness. scale is utilized as a tool for connecting “macro required performance
Because these ceramics utilize the oxidation reaction, the healing (e.g., strength, stiffness, and healing ability)” as products/components
behaviorissometimescalledoxidation-inducedcrackrepair[24].This and“microstructureconditionofmaterials.”
strategywasimplementedusingminiaturizedsiliconcarbide(SiC)[1–6, Inthecaseofself-healingmaterials,unlikeordinarymaterials,novel
19];titaniumcarbide(TiC)[13,15];intermetalliccompounds(TiSi2and numerical analysis methods to evaluate both the damage and healing
MoSi2)[9,11];morerecently,compoundedMAXphases[7,10,12–14,18, processes as reciprocally continuous processes need to be developed.
23];andothermaterials[8,16,17].Recently,oneoftheauthorseluci- Undersuchcircumstances,recentdevelopmentshavebeenobservedin
datedthattheself-healingmechanismofceramicsisachievedthrough formulatingbothdamageandhealingprocesseswithintheframeworks
* Correspondingauthor.
E-mailaddress:[email protected](S.Ozaki).
https://doi.org/10.1016/j.oceram.2021.100135
Received7March2021;Receivedinrevisedform29April2021;Accepted24May2021
Availableonline29May2021
2666-5395/©2021TheAuthor(s).PublishedbyElsevierLtdonbehalfofEuropeanCeramicSociety.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
S.Ozakietal. OpenCeramics6(2021)100135
ofcontinuumdamagemechanics[31–38]andcohesive-zonemodeling criterion.
[39]. In particular, the authors have proposed a continuum damage
model that incorporates the strength recovery behavior of self-healing 2.2. Oxidationkineticsmodelofcrackhealing
ceramics[40–42].Furthermore,theproposedmodelwasimplemented
usingthefiniteelementmethod(FEM)todemonstratenumericalanal- Next,theoxidationkineticsmodelofcrackhealingisoutlined.Osada
ysis of alumina/SiC particle composites under various conditions of etal.[43]succeededinmodelingthestrengthrecoverybythefollowing
temperatureandoxygenpartialpressure. passive oxidation, based on a three-dimensional observation of the
However,empiricalevolutionlawswereusedtodescribethehealing strengthrecoverybehaviorofalumina/SiCcomposites:
behavior in previous constitutive models, and the effects of crack-
3
opening width, temperature, and oxygen partial pressure on the SiCþ O ðgÞ¼SiO ðsÞþCOðgÞ (4)
strengthrecoveryratewerenotproperlyreflected.Theimportanceofthe 2 2 2
oxidation-kinetic or thermokinetic aspects of strength recovery was Specifically,theprocessofcrack-gapfillingcausedbytheoxidationof
pointedoutbyGreil[24].Inaddition,attemptstomodeltheweight/- SiCwasformulatedbasedon kinetics.Fig.1showsschematicillustra-
volume increase of oxides that fill cracks have already been reported tions of oxidation-induced crack-gap filling models in an alumina/SiC
basedonoxidationkinetics.Forexample,Osadaetal.[43]proposeda compositeandmonolithicSiC.
kineticsmodelthatreflectsthemicrostructuralfeaturesofthefracture Thevolumegainbyoxidecanbeexpressedasafunctionofhealing
surface,shapeofcracks,andtypeandamountofthehealingagent. temperature,healingtime,andsoon.First,thefollowinggeneralformof
In this paper, we propose a damage-healing constitutive model of theisothermalweightgainΔwisconsidered:
ceramics,inwhichtheevolutionlawofthecrack-healingbehaviorbased (cid:6) (cid:7)
on oxidation kinetics is incorporated. The proposed model was imple- Δw¼ k pt h 1=n (5)
mented in the FEM, and the damage-healing processes analyzed.
Furthermore, to verify the effectiveness of the proposed model and wherekpistherateconstant,thistheoxidationtime(healingtime),andn
analysis scheme, a comparison with experimental data reported for indicatestherate-controllingoxidationmechanism(n¼1,2forreaction-
Al2O3/15-vol.% SiC particles [5,6], Al2O3/30-vol.% SiC particles [25, anddiffusion-controlledweightgains,respectively).ForSiCoxidation,n
43],andmonolithicSiC[43]wasperformed. ¼ 2 can be adopted, and Eq. p(5 ffiffiffi) ffiffiffiffic ffiorresponds to Wagner's parabolic
weightgainmodel,i.e.,Δw ¼ k t .
ph
2. Constitutivemodel TherateconstantkpshowsatemperaturedependencyofArrhenius
type,i.e.,
In this section, the formulation of the isotropic damage and self- (cid:1) (cid:3) ! m
h noea mli en ng ac ao rn esti dtu esti cv re ibm edod be al seis dex op nla cin oe nd ti. nT uh ue mda dm amag ae gean md eh chea al nin icg sp ah ne d- k p¼k poexp (cid:2) RQ Tox P PO o2 (6)
h O2
oxidationkinetics,respectively.
wherekoandQ arethefrequencyfactorandtheactivationenergyfor
p ox
2.1. Damagemodel theoxidationofthehealingagent(SiC),respectively;Thisthehealing
temperature;andRisthegasconstant.Furthermore,inEq.(6),thein-
The stress–strain relationship based on the conventional isotropic fluenceoftheoxygenpartialpressureP O2 onkpisalsoconsidered.Here,
damagemodelisgivenasfollows: Po isthe standardoxygenpartialpressure,andmis thetemperature-
O2
independent constant. Goto et al. reported m ¼ 0.08–0.13 and
σ¼ð1(cid:2)DÞc:ε (1) 0.37–0.53forO2–ArandCO2–Aratmospheres,respectively[46,47].
From the abovementioned information, when an oxide film grows
whereσ,ε,andcaretheCauchystresstensor,smallstraintensor,and
one-dimensionallyfromtheSiCsurface(Fig.1),theweightgaincanbe
fourth-order elastic coefficient tensor, respectively. The scalar value D
described as a function of the oxidation temperature, oxygen partial
ð0(cid:3)D(cid:3)1Þisthedamagevariable,andD¼0andD¼1correspondtothe
undamagedstateandperfectlydamagedstate,respectively.()standsfor
pressure,andtime.Meanwhile,toestimatethevolumegainVhduetothe
oxidation,thisweightgainshouldbeconvertedintothevolumegainas
thesecond-orderscalarproduct.
follows:
Toincorporatethedamageprocessofbrittleceramics,thefollowing
c 3o 0h ,4e 0si –v 4e 2-z ,4o 4n ]e :relationisembeddedintotheisotropicdamagemodel[29,
V
h¼2A Δf
V
ρf
EΔw (7)
(cid:1) (cid:3)
σ¼σ texp (cid:2)
Gσ
t fw (2)
fracH tie or ne, oA fthis et Sh ie Ca pre aa rto icf leo ,n ae ns did fEe io sf tt hh ee ef fr fa ec ct tu ivr ee rs eu ar cfa tic ve e, afV reis at rh ae tiovo ol fu Sm iCe
onthefracturesurface,whichvarieswithin0.5–1.0.Theweightgainper
whereσisthecohesiveforceperunitfracturesurfaceandwisthecrack- unit volume gain, Δρ, can be calculated using the following equation
opening width. Further, σ t is the fracture stress, and Gf is the fracture whenthehealingagentisSiC:
energy.ByarrangingEq.(2)intheformofEq.(1),thedamagevariableis
M (cid:2)M
expressedasfollows[44]: Δρ¼(cid:1)SiO2 Si(cid:3)C (8)
D¼1(cid:2)κ 0exp(cid:4) (cid:2)σ th
eðκ(cid:2)κ
Þ(cid:5)
(3)
M ρ SS ii OO 22(cid:2)M ρ SS ii CC
κ G 0
f
where M and M are the molar massesand ρ and ρ are the
Here,κcanbeconsideredastatevariable.Intheconventionaldam- densitiesS oiO f2 SiO2andSiC
SiC,respectively.
SiO2 SiC
a vg ae luem oo fd te hl, et eo qd ue ivs acr leib ne tst th re aid na εm eqa ig ne th hi est lo or ay did ne gp hen isd toen rycy is, t ah de opm tea dxi fm orum
κ.
giveU nsi bn ygEqs.(5)–(7),thevolumegainVhduringisothermaloxidationis
c tF hhu eart rh fiae c nr t ie, tr eκ is0 eti li c es mlt eh ene ng tte hq au n(wi av lha yil sce ih snt c (o Fst Err r Aa ei )sn )poa [n 4t d 5d s ]a .tm o Ina thge te hli ien snit sgi tta uht dio o yn f ,, t rha een gad e rle dh m ie nei gs nt tt hh in ee
V
h¼2A Δf
V
ρf E(cid:6)
k pt
h(cid:7) 1=n¼2A Δf
V
ρf E(
k
poexp(cid:1) (cid:2) RQ Tox(cid:3) P
PO
o2! m
t
h) 1=n
(9)
fracture stress σ t, the maximum principal stress is adopted as the h O2
2
S.Ozakietal. OpenCeramics6(2021)100135
Fig.1. Illustrationsofoxidation-inducedcrack-gapfillingmodelsinanalumina/SiCcompositeandmonolithicSiC.
Finally,therateformoftheoxidationkineticsmodelsofthevolume Conversely, the self-healing part κ is assumed to be a monotonic
h
gainisgivenby increasing function, which describes the process κ→κ . Based on the
0
(cid:1) (cid:3) kinetics model of Eq. (10) and the oxidation-induced crack-gap filling
V_ h¼ nV1 n(cid:2)1 2A Δf V ρf E n k p (10) modelsshowninFig.1,theevolutionlawofκ hisgivenasfollows:
h 9
gw ih vee nre bV y_ histhevolumegainrate.Moreover,theinitialvolumegainV hois κκ __ h h¼
¼0AV_
hh e forf Oor thκ e> rwκ is0 e;
:
>>=
>>; (15)
(cid:4)(cid:1) (cid:3) (cid:5)
V ho¼
2A Δf
V
ρf
E
n
k pΔt
1=n
(11) Furthermore, to describe the dependence of the strain at damage
initiation after healing on the degree of self-healing, the maximum
whereΔtisthetimeintervalinthenumericalcalculation.NotethatEq.
equivalentstrainκreceivedinthepastmustbere-evaluated.Fortheself-
healingstate,weassumethatthemaximumequivalentstraingradually
(10) can be applied to not only isothermal condition but also non-
approachesκ, andthustheevolutionlawofthemaximumequivalent
isothermalconditionsbysimultaneouslyusingitwithEq.(6). s
straincanbegivenasfollows:
2.3. Evolutionlawsforstatevariables κ_¼(cid:2)ξκ_ hsgnðκ(cid:2)κ sÞþκ_ ε (16)
In this section, we briefly explain how the evolution laws for self- Here,sgn()representsthesignfunction.Inthereloadingstateafter
healingareincorporatedintotheconstitutivemodel.
healing,damagedoesnotoccuruntilε
eq
¼κ.Here,ξistheparameter
Toincorporatetheself-healingbehaviorintothedamagemodelofEq. affectingthestrengthrecoveryrate.Notethatwhenκ s>κ 0,thefracture
(1),thedamagevariableisassumedtoevolvefromthestateofD6¼0 strengthofthehealedpartbecomeshigherthanthatoftheundamaged
towardsD¼0throughself-healing,dependingonthetemperatureand part. Thisphenomenonis causednot onlyby the strengtheningof the
theoxygenpartialpressure.Inotherwords,thedamagehistorydisap- healedpartbutalsobythescatterofthestrengthduetointernalpores.
pearswithself-healing,andthestatevariableκevolvestothesoundness Therefore,κ s>κ 0indicatesthattheinternalporethatistheoriginofthe
state.Basedonthisconcept,weassumedthatthestatevariableκaddi- initialcrackisalsofilledbytheoxide;hence,thefracturestrengthafter
tivelydecomposedintotheequivalentstrain(damage)partκ ε andthe healingdependsonthespatialdistributioncharacteristicsofperipheral
self-healingpart(cid:2)κ asfollows: pores[42].
h
Thedamageandtheloadingcriteriaaredescribedby
κ¼κ ε(cid:2)κ h (12) 9
ifκ<κ →D¼0 forundamaged; =
0
where the evolution of the damaged part is given by the following ifκ<κ →D¼DðκÞforκ(cid:4)κ 0; ; (17)
equation:
ifκ¼κ →D¼DðκÞfordamageprogress:
κ_ ε¼〈ε_ eq〉 (13)
〈ε_ eq〉>0: loading;
9 >>=
damH ae gr ee, p< ro> gred se sn .o Ste ims iM lac rA tu oley p' rs evb ir oa uc ske st ts u. dκ iε esev [o 2lv 9e ,3s 0o ,4n 0ly –4d 2u ,4ri 4n ]g
,
t whe
e
κ〈 _ε h_ eq >〉¼ 0:0: heau ln inlo ga :dingorholding; >>; (18)
adoptedthefollowingmodifiedvonMisesequivalentstrainε ,whichisa
eq
scalarvalue,fortheevolutionofdamage: Notethattheinitialvalueofκfortheundamagedstatecorrespondsto
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ .
(cid:1) (cid:3) 0
k(cid:2)1 1 k(cid:2)1 2 12k
ε eq¼ 2kð1(cid:2)2νÞI 1þ
2k
1(cid:2)2νI
1
þ ð1þνÞ2J
2
(14)
2.4. Responsecharacteristicsofthedamage-healingconstitutivemodel
where ν is the Poisson ratio, k is the ratio of tensile and compressive Fig. 2 shows schematic response characteristics of the damage-
strengths,I1isthefirstinvariantofthestraintensor,andJ2isthesecond healingprocessesbythepresentconstitutivemodelduring(i)uniaxial
invariantofthedeviatoricstraintensor. tensionloadingandunloading,(ii)healing,and(iii)reloading.Here,the
3
S.Ozakietal. OpenCeramics6(2021)100135
Fig.2. Schematicresponsecharacteristicsofdamage-healingprocessesobtainedbythepresentconstitutivemodelduring(i)uniaxialtensionloadingandunloading,
(ii)healing,and(iii)reloading.
upperpartofthefigurepresentsthestress–timerelation,whilethelower (1) Alumina matrix composite containing 15 vol% SiC particles
partshowsthevariationsintherelevantstatevariables. (AS15P)
Asshownintheloadingstage(i)inthefigures,theconstitutivemodel (2) Alumina matrix composite containing 30 vol% SiC particles
showselasticbehavioruntilthemaximumprincipalstressreachesσ tand (AS30P)
transitions to D6¼0 state with crack initiation. Simultaneously, the (3) CommercialmonolithicSiC(SC-211;KyoceraCo.Ltd)(SiC100)
equivalentstrainatdamageinitiationκ isevaluated.Next,thestiffness
0
decreases according to the magnitude of D. After the initiation of Rectangularbarsofspecimenswithdimensions3mm(cid:5)4mm(cid:5)22
cracking,theoxidationreactionstarts,andthecrackfillswithoxidesin mmwereused,whichwerepolishedtoamirrorfinish.
thehealingstage(ii).Atthesametime,theequivalentstrainκevolves Testspecimenswithnodamagearecalled“undamagedspecimens.”
dependingonthetemperatureandtheoxygenpartialpressure,andthe Toevaluatethehealingperformance,theprescribeddamagewasinflic-
damagevariablerecoverstoD¼0.Inaddition,themaximumequivalent tedtotheundamagedspecimens.Inparticular,asemi-ellipticalsurface
strainκevolvestowardsκ inresponseasthecrackedpartissufficiently pre-crackwithalengthofalmost100μmandanaspectratioof0.9were
s
healed.Inthesubsequentreloadingstage(iii),crackingoccursatε ¼ κ. introduced at the center of each undamaged specimen surface by the
eq
Thus,thedependencyofstrengthrecoveryonthedegreeofself-healingis Vickersindentation.Thesepre-crackedspecimensarecalled“as-cracked
naturallydescribed.Notethatduringthedamageprocessinstages(i)and specimens.” The as-cracked specimens were healed under the given
(iii),self-healingalsooccurs.However,forthesakeofbrevity,thevari- conditionsofthehealingtemperaturebetween700and1350(cid:6)C,healing
ationsinthestatevariablesareignoredinFig.2. timebetween1minand100h,andoxygenpartialpressurebetween50
and 21000 Pa. These specimens are called “crack-healed specimens.”
3. Analysismodelandconditions Here,the heatingand coolingtemperature ratesin the heattreatment
processduringhealingwere10(cid:6)C/minand5(cid:6)C/min,respectively.The
Inthisstudy,athree-pointbendingtestwasadoptedastheanalysis temperatureconditionsduringhealingareshowninFig.3.
target with reference to previous experiments [6,43] to verify the The strength recovery of each specimen was measured by a three-
effectivenessoftheproposedkinetics-basedconstitutivemodel.Here,the point bending test at room temperature (22 (cid:6)C), in which the span
strengthofbrittleceramicsstronglydependsontheeffectivevolumeof lengthwas16mm.Inthistest,thestrengthrecoverywasevaluatedby
thebendingsystem[30].Thus,athree-pointbendingtesthasasmaller settingthepre-crackedorhealingpartsonthesurfaceonthetensileside.
effective volume than a four-point bending test, and enables us to Fordetailsonthepreparationofthetestspecimensandtheexperi-
conductaccuratesafety-sideevaluationsofthestrengthrecovery.Inthis mentalprocedure,refertoRefs.[6,43].
section,thetargetexperimentandspecimens,correspondingFEAmodel,
andconditionsareoutlined. 3.2. Finiteelementmodelandanalysisconditions
IntheFEA,weusedthecommercialsoftwarepackageLS-DYNAand
3.1. Targetexperiment itsrelatedusersubroutine[48].Fortheanalysisofthedamageprocess,
thedynamicexplicitmethodbasedonthecentraldifferencemethodwas
The following three types of materials were used in the target adopted, and for the analysis of the self-healing process, the dynamic
experiments: implicit method based on the time discretization of the Newmark β
4
S.Ozakietal. OpenCeramics6(2021)100135
thedisplacementsofthelowerjigswerefixed.Africtioncoefficientof0.1
wassetatthecontactboundarybetweenthespecimenandthejigs.Here,
the jigs of the three-point bending test were assumed as rigid bodies.
Note that the stress, strain, and state variables in each element were
inheritedbetweenthehealinganddamageanalyses,andviceversa.
3.3. Inputparameters
Tables1and2showthemechanicalpropertiesandself-healingpa-
rametersofeachspecimen,respectively.Thesevalueswereadoptedby
referring to previous studies [6,43]. In this study, Young's moduli of
alumina/SiC composites were evaluated using the composite law.
Further,thefractureenergywasevaluatedusingthefollowingequation:
K2 (cid:6) (cid:7)
Fig.3. Temperatureconditionsduringhealinginbothexperiment[6,43]and G f¼ EIC 1(cid:2)ν2 (19)
finiteelementanalysis.
whereK isthefracturetoughnessofmodeI.Inaddition,thefracture
methodwasadopted.TherestartfunctionofLS(cid:2)DYNAwasutilizedfor IC
stress was determined from the scale parameter of the Weibull distri-
connectingthehealingandloadingstagesbecausethetimescalesofthe
butionofthesamematerial.
loading/unloadingstageswereverydifferentfromthoseofthehealing Meanwhile,theinitialequivalentstrainκ ofthepre-crackedpartis
stage.ThespecificationsoftheFEAmodelandconditionsaredescribed d
relatedtotheaveragevalueofthecrack-openingwidthwbytheVickers
below.
indentationasfollows:
Fig.4showstheFEAmodelforthethree-pointbendingtest,wherea
one-pointGaussianintegrationwasadopted.Further,Fig.5showsthe w¼h eðκ d(cid:2)κ 0Þ (20)
detailsofmodelingoftheinitialdamagedpart,whichcorrespondstothe
IntheFEA,κ wasapproximatedbythefollowingequationusingthe
pre-crackbytheVickersindentation.Notethatintheevaluationofthe d
fracturestressσ:
crack-healedspecimensbytheFEA,theVickersindentationprocesswas t
n (do at mp ae gr efo drm pae rd t). wTh itu hs, at lh ee nga ts h-c or fac 2k ce ¼d 1sp 0e 0ci μm men ws erw eit ph ret ph ae rei dn ,it aia nl dc tr ha ec ik
r
κ d(cid:7)σ Etþ hw (21)
e
healinganalysiswasperformed.Subsequently,thethree-pointbending
analyses were performed.Here, only the directionof the cross-shaped wherew¼60nm.Noteagainthattheaveragevalueobtainedfromthe
crack that affected the fracture behavior was considered, and a semi- experimentwasusedforthecrack-openingwidthw.Thatis,thedamage
elliptical crack of one element in width was modeled (Fig. 5). To ex- amounts of each element in the initial damaged part in Fig. 5 are the
pressthestressconcentration,thecentralpartofthespecimenmodelwas same.Forthesakeofsimplicity,wesetκ s¼κ 0andξ¼1inthefollowing
discretizedwithafinecubicmesh(he¼0.025mm),andtheproposed FEA.
damage-healing constitutive model was applied only there. Thus, the TheeffectivereactivearearatioofSiConthefracturesurface,fE,was
areaofonesideofthefracturesurface,A,isgivenbyh2 intheconsti- setto1.0formonolithicSiC,whereasitwassetto0.5foralumina/SiC
e
tutivemodel.Meanwhile,thesurroundingpartwasalinearelasticbody compositesbecauseSiCparticlesarelocatedononesideofthefracture
withthesameelasticcharacteristicsasinthedamage-healingmodel.The surface(seeFig.1)[43].
damagemagnitude(initialvalueofthedamagevariableD)attheinitial
damagedelementswasevaluatedfromtheaveragevalueofthecrack- 4. FEAresultsanddiscussion
openingwidthwbasedontheexperimentalobservation[6,43].
Intheself-healingprocess,jigswereignored,andthedisplacement 4.1. Resultsofthehealinganddamagebehaviors
conditionofthespecimenswascompletelyfixed.Next,theself-healing
process under the given temperature and oxygen partial pressure con- First,wedescribethetypicalresultsofthehealinganddamagebe-
ditionswasanalyzed,asshowninFig.3.Incontrast,inthebendingtest haviorsusingtheproposedconstitutivemodel.Figs.6and7showthe
calculations, aconstant forcedvelocity(constantcrossheadspeedof 5 timeseriesvariationsofthesnapshotsofthedamagevariabledistribu-
mm/s)intheverticaldirectionwasimposedontheupperjig,whereas tion during healing in the as-cracked specimens of AS30P. Here, the
figuresshowaclose-upviewaroundtheinitialdamagedpart.Theblue
elementsrepresentthestatewithoutdamage(D¼0),andtheredele-
mentsrepresentthedamagedstate(D(cid:7)0.6).IntheFEA,wesetthesame
conditionsofthetemperatureThandoxygenpartialpressureP O2asinthe
experiment[43].Notethatthehealingtimeintheresultsdescribedin
Section4correspondstothesteady-state(holding)timeinFig.3.
Fig.6showstheeffectofthehealingtemperatureunderaconstant
oxygenpartialpressurecondition.Becausetheevolutionlawofthestate
variableκ (Eq.(15))isintroduced,itcanbeconfirmedthatthedamaged
h
part heals to a soundness state within the healing time. Furthermore,
because the proposed constitutive model is formulated based on the
oxidationkineticsofEq.(10),itcanalsobeconfirmedthatthehealing
ratechangesaccordingtothehealingtemperature.Inaddition,asshown
inFig.7,theeffectoftheoxygenpartialpressureissimultaneouslyre-
flected.Iftheoxygenpressureistoolow,itwillhardlyhealevenathigh
temperatures.
Fig.4. Finiteelementanalysismodelofathree-pointbendingtest. Fig.8showsthedamageprogressbehaviorbythree-pointbending.
5
S.Ozakietal. OpenCeramics6(2021)100135
Fig.5. Detailsofmodelingoftheinitialdamagedpartcorrespondingtoapre-crackbytheVickersindentation.Asemi-ellipticalcrackisapproximatedbysixcubic
elementswithasideof0.025mm,whereastheaveragecrack-openingwidthis60nm:(a)overallviewand(b)detailedview.
Table1 Table2
Mechanicalproperties. Parametersofself-healing.
E[GPa] ν σ t[MPa] KIC[MPa√m] k ko p½kg2=ðm4sÞ(cid:8) Qox[kJ/mol] R[J/(molK)] m fE
AS15P 389 0.23 891 4.0 10 7.85(cid:5)10(cid:2)7 161 8.314 0.835 0.5,1.0
AS30P 398 0.22 979 4.0 10
SiC100 440 0.16 606 7.0 10 fV MSiO2½g=mol(cid:8) MSiC½g=mol(cid:8) ρ SiO2½kg=m3(cid:8) ρ SiC½kg=m3(cid:8)
0.15,0.3,1.0 60.1 40.1 2320 3210
ThefiguredepictsthecontourmapofthedamagevariableD,andthered
part shows the damaged part (D (cid:7) 1.0). In addition, Fig. 9 shows the
relationship between the bending stress and the deflection of each
6
S.Ozakietal. OpenCeramics6(2021)100135
Fig.6. Timeseriessnapshotsofthehealingprocessoftheinitialdamagedpartinanas-crackedspecimenofAS30P.Here,thetemperatureconditionsare1100,1200,
and1300(cid:6)C,andtheoxygenpartialpressureis21,000Pa.Theblueelementsrepresentthestatewithoutdamage,andtheredelementsrepresentthedamagedstate.
(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtotheWebversionofthisarticle.)
Fig.7. TimeseriessnapshotsofthedistributionofthedamagevariableDoftheinitialdamagedpartduringthehealingprocessinas-crackedspecimensofAS30P.
Here,thetemperatureconditionis1350(cid:6)C,andtheoxygenpartialpressuresare50,5000,and21,000Pa.
specimen.Fromthefigures,itcanbeobservedthatthebendingstress the bending stress–deflection relationship, although the as-cracked
increases linearly with deflection. The damage progresses when an specimen has lower strength than the undamaged specimen, the
elementonthebottomcenterofthespecimenreachesthefracturestress, strengthofthecracked-healedspecimenisclearlylargerthanthatofthe
andthenthebendingstresssharplydecreases.Hence,thepresentFEA as-crackedspecimen.Astheseresultsconfirm,thefracturestrengthofthe
modelcanrepresentthetypicalbrittlefracturebehaviorofceramics.In initialdamagedpartisrecoveredduetothehealingeffect.Itshouldbe
addition,theadvantageofusingtheFEAisthattheeffectofstressdis- noted that the crack-healed specimen exhibits a complete strength re-
tributionaroundthedamagedpartcanbetakenintoconsiderationfor coveryifthehealingtimeissufficientlylong(seethenextsection).
thebendingstrength.Thestressconcentrationdependingonthehealing
stateisnaturallyexpressedandreflectedintheresultsshowninFigs.8
4.2. Comparisonwithexperiments
and9.
Fig. 9 shows the results of the three-point bending test for the un-
Inthissection,aquantitativecomparisonbetweentheFEAresultsand
damaged,as-cracked,andcrack-healedspecimens.Asdemonstratedby
theexperimentalresults[6,43]ispresented.
7
S.Ozakietal. OpenCeramics6(2021)100135
Fig.8. Finiteelementanalysisresultsofthedamagepropagationduetothree-
pointbending,withthecontourmapofthedamagevariable.
Fig.10. TimedependenceofthestrengthrecoveryofAS15Pobtainedbythree-
point bending in the finite element analysis and experiment [6]. The plots
correspondtothepeakvaluesofthebendingstress–deflectionrelationships:(a)
resultsoffourlevelsoftemperatureunderaconstantoxygenpartialpressure
condition;(b)resultsofthreelevelsofoxygenpartialpressureunderonetem-
peraturecondition.
recovery magnitude was obtained by the three-point bending analysis
after the healing process, as in the experiment. Separate plots of the
graph show the undamaged, as-cracked, and crack-healed specimens.
Theemptytriangles(i.e.,△)correspondtotheexperimentalvalues,and
Fig.9. Relationshipbetweenthebendingstressandthedeflectionobtainedby thefilledcircles(i.e.,●)andthedottedlinecorrespondtotheFEAre-
thefiniteelementanalysisofthreespecimens,i.e.,as-cracked,crack-healed,and
sults.First,althoughthebendingstrengthoftheas-crackedspecimenis
undamagedspecimens.
higherthanthatintheFEAowingtotheeffectofthediscretizationsizeof
the elements, the present FEA model can reasonably represent the
Fig. 10 shows the results of the FEA and experiment on the time
bending stress in both the as-cracked and undamaged specimens.
dependenceofstrengthrecoveryinAS15P.Fig.10(a)showstheresults
Furthermore,asshowninFigs.6and7,theinitialdamagedpartheals
underfourtemperatureconditionsandaconstantoxygenpartialpressure
withintheelapsedtimeandchangestoasoundnessstate.Basedonthese
condition,andFig.10(b)showstheresultsunderthreeoxygenpartial
results,theFEAresultsdemonstratethatthestrengthoftheas-cracked
pressure conditions and one temperature condition. The strength
specimen recovers as the healing time is increased and reaches the
8
S.Ozakietal. OpenCeramics6(2021)100135
same level as that of the undamaged specimen, as in the experiment.
Notably,thetimerequiredforcompletehealingundereachconditionis
in reasonably good agreement with the experimental value. Here, the
reasonwhytheinitialvalueoftherecoveredstrengthdiffersdepending
ontheself-healingtemperatureisthatthehorizontalaxesofthegraphs
aretheholdingtimeinaconstanttemperaturesection.Infact,healing
alsooccursintheincreasing-anddecreasing-temperaturesections(see
Fig.3);thus,thedifferenceoccurs.
Toexaminetheeffectivenessoftheproposedconstitutivemodelfor
thechangesinthecompositeratiooftheself-healingagent(SiC),wealso
comparedthetimedependencesofthestrengthrecoveryinAS30Pand
SiC100.
Fig. 11 shows the results of the FEA and experiment on the time
dependenceofthestrengthrecoveryinAS30P.Here,theresultsunder
four temperature conditions and a constant oxygen partial pressure
condition are presented. Although the FEA results at 1300 (cid:6)C show a
slightlyslowerrecoveryratethantheexperimentalresult,itcanbeseen
thattheexperimentalresultscanbereproducedwellbytheFEA,asinthe
caseofAS15P.Moreover,itisalsoconfirmedthatthehealingratebe-
comesfasterthanthatofAS15P,reflectingtheSiCcompositeratio.
Fig.12showsacomparisonofthetemperaturedependencesofthe
strengthrecoveryinSiC100betweentheFEAandexperiment,wherethe Fig.11. TimedependenceofthestrengthrecoveryofAS30Pobtainedbythree-
healingtimethis1hunderatmosphericpressure.Althoughtheexperi- point bending in the finite element analysis and experiment [43]. The plots
mentalresultisslightlyfasterthantheFEAresult,theoveralltendency correspond to the peak values of the bending stress–deflection relationships.
can be predicted. The healing rate obtained using the FEA is slightly Healing conditions are four levels of temperature under a constant oxygen
slowerthantheexperimentalonebecausetheaveragevalueofthecrack- partialpressure.
opening width w (corresponding to the initial damaged value) by the
VickersindentationisthesameasthatofAS15PandAS30P.Thepre-
dictionresultsmaybeimprovedbymatchingtheinitialcrack-opening
widthtothatofSiC1000.
Fromtheinformationabove,thebasiccharacteristicsofstrengthre-
covery inself-healingceramicscan beobtainedusingthe presentFEA
methodology implementing the damage-healing constitutive model.
However,intheexperiment,thescatterofstrengthwasobservedinboth
theundamagedandcompletelycrack-healedspecimens.Thestrengthof
ceramicsvariesduetothedistributioncharacteristicsofinternaldefects.
Therefore,evenwhentheinitialdamagedpartiscompletelyhealed,the
fractureoriginmovestothenextweakestpartaroundthebottomcenter.
Thus,thebendingstrengthofthecrack-healedspecimensalsoshowsa
scatter of strength depending on the distribution characteristics of in-
ternaldefectsintheperipheralregionofthehealedpart.Regardingthis
point,wehavealreadyproposedamethodforpredictingthescatterof
strengthbasedonthedistributioncharacteristicsofinternaldefects[29,
30] and have confirmed that the problem can be handled by using
damage-healinganalysis[42].
5. Conclusions
Inthisstudy,weproposedadamage-healingconstitutivemodelfor
Fig.12. Comparisonofthetemperaturedependencesofthestrengthrecovery
self-healing ceramics, in which the evolution law of crack-healing inSiC100betweenthefiniteelementanalysisandexperiment[43],wherethe
behavior based on oxidation kinetics was incorporated. Subsequently, healingtimethis1hunderatmosphericpressurecondition.Theplotscorre-
theproposedmodelwasimplementedintheFEM,andaseriesofdamage spondtothepeakvaluesofthebendingstress–deflectionrelationships.
and healing processes of alumina/SiC composites was performed.
Furthermore, the effectiveness of the proposed model and analysis paves the way for a simulation of crack-healing behavior for the non-
scheme was demonstrated using a three-point bending test analysis. prescribed crack, which is calculated by prior damage analysis under
Thereafter, the time dependence of the strength recovery obtained arbitraryboundaryconditions.Meanwhile,wehavealreadyproposedan
throughtheFEAwasquantitativelycomparedwiththeexperimentaldata FEAmethodologytoevaluatethescatterofstrengthinceramics[29,30,
reportedforAl2O3/15-vol.%SiCparticles,Al2O3/30-vol.%SiCparticles, 42].Wewillworkonconcreteissuesbyusingthesenumericalschemesin
andmonolithicSiC.ItwasconfirmedthatthepresentFEAmethodology
thenearfuture.
couldreasonablyreproducethebasiccharacteristicsofstrengthrecovery Finally, it is must be mentioned that we focused on the fracture
inself-healingceramics.Thus,weconcludethatthepresentFEAmeth- processcausedbyonlythetensilecrackingduetothree-pointbending.
odologycanbeusedforstudyingtheself-healingbehaviorlinkedwith TheeffectivenessofthepresentconstitutivemodelformodeIIandmode
themicrostructuredistributionandfractureproperties,whichisessential IIIcrackingandcompressivefracturesmustbealsoexamined.Further-
for the mechanical and material design of self-healing ceramics. more,theeffectivenessofthepresentFEAforceramicmatrixcomposites
Althoughthisstudyfocusedonthestrengthrecoveryfortheprescribed havingarbitraryshapesundermorecomplexboundaryconditionsshould
pre-crackintroduced by the Vickersindentation, the proposed scheme
9
S.Ozakietal. OpenCeramics6(2021)100135
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healingceramicsformiddlerangetemperature,KeyEng.Mater.810(2019)
ThisworkwassupportedbyaGrant-in-AidforScientificResearch, 119–124.https://doi.org/10.4028/www.scientific.net/KEM.810.119.
JSPS, Japan [grant numbers (C) 19K04088 and (B) 19H02033]; and
[20] CF. eF r. aL ma .n Sg oe, c.K 5.C 3. (R 1a 9d 7f 0o )rd 4, 2H 0e –a 4l 2in 1g ,ho tf tpsu s:r /f /a dce oic .ora rgck /1s 0in .1p 1o 1l 1y /c jr .y 1s 1t 5al 1l -ineAl2O3,J.Am.
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Declarationofcompetinginterest [22] K.Ando,T.Ikeda,S.Sato,F.Yao,Y.Kobayasi,Apreliminarystudyoncrackhealing
behaviorofSi3N4/SiCcompositeceramics,Fatig.Fract.Eng.Mater.Struct.21
The authors declare that they have no known competing financial (1998)119–122.
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