A cohesive-zone crack healing model for self-healing materials

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A cohesive-zone crack healing model for self-healing materials

Ponnusami, Sathiskumar A.; Krishnasamy, Jayaprakash; Turteltaub, Sergio; van der Zwaag, Sybrand

DOI

10.1016/j.ijsolstr.2017.11.004

Publication date

2018

Document Version

Final published version

Published in

International Journal of Solids and Structures

Citation (APA)

Ponnusami, S. A., Krishnasamy, J., Turteltaub, S., & van der Zwaag, S. (2018). A cohesive-zone crack

healing model for self-healing materials. International Journal of Solids and Structures, 134, 249-263.

https://doi.org/10.1016/j.ijsolstr.2017.11.004

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ContentslistsavailableatScienceDirect

International

Journal

of

Solids

and

Structures

journalhomepage:www.elsevier.com/locate/ijsolstr

A

cohesive-zone

crack

healing

model

for

self-healing

materials

Sathiskumar

A. Ponnusami

1

_,

_Jayaprakash

_Krishnasamy,

_Sergio

_Turteltaub

∗

_,

_Sybrand

_van

_der

Zwaag

Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, Delft 2629 HS, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 7 July 2017 Revised 28 September 2017 Available online 7 November 2017 Keywords:

Self-healing material Cohesive-zone model Multiple crack healing Fracture mechanics

a

b

s

t

r

a

c

t

Acohesivezone-basedconstitutivemodel,originallydevelopedtomodelfracture,isextendedtoinclude ahealingvariabletosimulatecrackhealingprocessesandthusrecoveryofmechanicalproperties.The proposedcohesiverelationisacomposite-typematerialmodelthataccountsforthepropertiesofboth theoriginaland thehealingmaterial,whicharetypically different.The constitutivemodelisdesigned tocapture multiplehealingevents,whichisrelevantforself-healingmaterialsthatarecapableof gen-eratingrepeatedhealing.Themodelcanbeimplementedinaﬁniteelementframeworkthroughtheuse ofcohesiveelementsortheextendedﬁniteelementmethod(XFEM).Theresultingnumericalframework iscapableofmodelingbothextrinsicand intrinsicself-healingmaterials.Salientfeaturesofthemodel aredemonstratedthroughvarioushomogeneousdeformationsandhealingprocessesfollowedby appli-cationsofthemodeltoaself-healingmaterialsystembasedonembeddedhealingparticlesunder non-homogeneousdeformations.Itisshownthatthemodelissuitableforanalyzingandoptimizingexisting self-healingmaterialsorfordesigningnewself-healingmaterialswithimprovedlifetimecharacteristics basedonmultiplehealingevents.

1. Introduction

The concept of self-healing is a promising path to enhance the damage tolerance and extend the lifetime of structural and functional materials. Research in the last decade hasshown that

self-healing mechanisms can be incorporated in various

mate-rial classes ranging from polymers to high temperature

ceram-ics(vanderZwaag,2008).Aclassicalexampleofan extrinsic self-healing mechanism is based on healing capsules (particles) dis-persed within the base (ormatrix) material as shownin Fig.1a. Inanextrinsicsystem,thehealingprocessistypicallyactivatedby cracks thatinteractwiththecapsules(Whiteetal., 2001;Kessler etal., 2003;Carabat etal.,2015).Such capsule-based self-healing mechanismfindsitsapplicationinmaterialsthatotherwisedonot possessthecapacitytorepairdamage.Otherextrinsicself-healing materials involve hollow fibers filled with healing agents (Pang and Bond, 2005a; Dry, 1994) and microvascular networks with

∗ _{Corresponding author.}

E-mail addresses: S.AnusuyaPonnusami@tudelft.nl ,

sathis.ponnusami@eng.ox.ac.uk (S.A. Ponnusami), J.Krishnasamy-1@tudelft.nl (J. Kr- ishnasamy), S.R.Turteltaub@tudelft.nl (S. Turteltaub), S.vanderZwaag@tudelft.nl (S. van der Zwaag).

1 Current address: Solid Mechanics and Materials Engineering, Department of En-

gineering Science, University of Oxford, Parks Road, OX1 3PJ Oxford, United King- dom.

distributed healing agents (Toohey et al., 2007). Among the ex-trinsic self-healing mechanisms, the encapsulated particle-based systemhas been widely investigated (Blaiszik etal., 2010; Wik-tor and Jonkers, 2011; Van Tittelboom and De Belie, 2013). The encapsulation-basedhealing concept maylead toa desirable au-tonomousself-healingbehaviorforthesystem(basematerialplus healingagent),but itis typically limitedby the ﬁniteamount of thehealingagentwhichoftenprecludesmultiplehealing,atleast inthe locationwhere thehealingagenthasbeen consumed. Mi-crovascularnetwork-based self-healingsystems offerthe possibil-ityof multiple healingafter repeateddamage events by enhanc-ing the supply of the healing agent. In contrast with extrinsic self-healingmaterialsystems,inintrinsicself-healingmaterialsthe healingactionis dueto thephysio-chemical nature ofthe mate-rialitself,asillustratedinFig.1b(BergmanandWudl,2008).Such materialshavethenaturalcapabilityofrepairingthedamagemore thanonce(Sloofetal.,2016).

Thoughextensiveresearchhasbeenconductedinrealizingsuch materialsystemsexperimentally,effortstodevelopcomputational modelsthatcouldsimulatefractureandhealinghavebeenscarce. Simulation-baseddesign guidelinescan be usedto optimize self-healingsystems.Consequently,thegoal ofthepresentworkis to develop a computational framework to model the effectof crack healingbehavior on the mechanicalperformance ofthe material orthestructure underconsideration.The modelingand computa-https://doi.org/10.1016/j.ijsolstr.2017.11.004

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Fig. 1. Schematic of (a) a capsule-based extrinsic self-healing material: a matrix crack is attracted towards the healing capsule, which upon fracture releases the healing agent into the crack, resulting in crack healing and (b) an intrinsic self-healing material: the healing agent is available directly from the material chemical composition. The material that ﬁlls the crack is typically different than the matrix material for both the extrinsic and intrinsic cases.

tionalframeworkiskeptsuﬃcientlygeneralsuchthatitiscapable ofanalyzingbothextrinsicandintrinsicself-healingmaterials.

Inthecontextofacapsule(orﬁber)-basedextrinsicself-healing system,therearetwocriticalaspectsthatneedtobeaddressedin ordertoachievearobustself-healingsystem.Firstly,acrack initi-atedinthehost(ormatrix) materialshould beattractedtowards thehealingparticle(or ﬁber)andfurthershould breakthe parti-cleforhealingtooccur.A schematicofsuch self-healingmaterial andthe associatedhealing mechanismis shownin Fig.1. Crack-particleinteraction,which isa crucial aspectto successfully trig-gerthe healing mechanism, has been analyzed parametrically in Ponnusamietal.(2015a,2015b)to generatedesignguidelinesfor the selection of the healing particles in terms of their mechan-ical properties.Other studies in the literature have utilized ana-lyticalandnumericaltechniquestoinvestigatetheinteraction be-tween the crack and the healing particles or capsules (Zemskov etal., 2011; Gilabert etal., 2017; Šavija etal., 2016). The second criticalaspectina self-healingsystem,relevant forboth extrinsic andintrinsicmechanisms,iscenteredonhowthematerial recov-ersitsmechanicalpropertiesoncethehealingmechanismis acti-vatedinornearthefracturesurfaces.Inparticular,therecoveryof load-carryingcapabilityasafunctionofhealingparameters, crack lengthand capsule propertiesis a subjectofimportance but has notreceivedadequateattentionintheliteratureyet.Consequently, onemainfocusofthecurrentresearchistosimulatetherecovery ofmechanicalpropertiesoftheself-healingsystem.

Researcheffortshavebeenmadeintheliteraturetomodelthe mechanical behavior of materials taking both fracture and heal-ingintoaccount. Mostoftheexisting models adopta continuum

damagemechanics-based approachwherebycracking andhealing

areinterpreted asa degradation orrecovery of material stiffness and strength (Barbero et al., 2005; Voyiadjis et al., 2011; 2012; MergheimandSteinmann,2013;Darabietal.,2012;Xuetal.,2014; Ozakietal.,2016).Thecommonfeatureofthesemodelsisthatthe internalvariablesdescribingthecontinuum degradationand heal-ingofthe materialrefer tothe effectivebehavior of(unresolved) crackingandhealingevents.Correspondingly,detailsatthelevelof individualcracksarenot explicitlytakenintoaccount.However,a directdescriptionatthelevelofindividualcracksandhealing par-ticlesisdesirableinviewofdesigningorﬁne-tuningaself-healing material.

Cohesive zone-based approacheshave also been proposed for

modelingcrackhealing.Unlikecontinuumdamagemodels,the ad-vantageoftheabovecohesivezoneapproachesisthatthematerial damageistreatedinadiscrete mannerascohesivecracks, which allowsforexplicitmodeling ofcrackevolutionandits healing. In MaitiandGeubelle(2006),crackhealingissimulated throughan artiﬁcialcrackclosuretechniquebyintroducing a wedgeintothe crack.Themethodologyisimplementedinaﬁniteelement

frame-workusingcohesiveelementsforsimulatingfractureandacontact lawthat enforces theconditions forcrack healingorretardation. InSchimmelandRemmers (2006)aMode Iexponential cohesive zonemodelisproposedtosimulatecrackhealingbyintroducinga jumpincrackopeningdisplacement.Aftermodelveriﬁcation,they applied theframework to simulatedelamination crack healingin

a slender beam specimen to show the capability of the model.

Some limitations of the model are with regard to the multiple

healingeventsandtheirone-dimensionality.In Uraletal.(2009),

a cohesive zone model for fatigue crack growth is developed,

which alsoconsiders crack retardationduringunloading regimes. InAlsheghriandAl-Rub(2015) athermodynamics-based cohesive

zone methodology is used to model crack healing behavior by

extending previous work on continuum damage-healing

mechan-ics(Darabi etal., 2012). The model takes intoaccount theeffect ofvariousparameterssuch astemperature,restingtimeandcrack closureon thehealing behavior. However, thefracture properties ofthe healedzone,upon complete healing, assume thevaluesof theoriginal material,which isoftennot thecaseevenfor intrin-sicself-healingmaterials.Furthermore,thecapabilityofsimulating multiplehealingeventsisnotdemonstratedinmanyofthe above-mentioned studies, whichis ofdirect relevance forintrinsic self-healingmaterialsorextrinsicsystemswithacontinuoussupplyof healingagent. Other modelingapproaches explored in the litera-turetosimulatecrackhealingbehaviorarebasedonthetheoryof porousmedia(Bluhmetal.,2015)orthediscreteelementmethod (LudingandSuiker,2008;HerbstandLuding,2008).

Toovercomethelimitationsintheexistingmodelsinthe litera-ture,ageneralizedcohesive-zonebasedcrackhealingmodelis de-velopedhere, whichcanbe appliedtobothextrinsicandintrinsic self-healingmaterials.Themodeliscapableofsimulatingproperty recovery after multiplehealing eventsand isalso able tohandle differentfracture propertiesforthe healingmaterialascompared tothatoftheoriginalmaterial.Anadditionalfeatureofthemodel isthatthepropertiesofthehealingmaterialmaybespecified sep-arately fordifferent healing instances. This is particularly impor-tantastherecovery ofthefracture propertiesinthehealedzone isnotalwayscomplete,resultinginvaryingfracturepropertiesfor eachhealinginstancethatdepend,amongothers,onhealingtime, diffusion-reactioncharacteristicsandtemperature.It isnotedthat the modeldeveloped heredoesnot explicitlyaim to capturethe actual healingkinetics, butto simulatethe recovery of the over-allloadbearingcapacityasafunctionofcrackfillingandfracture propertiesofthefillingmaterial.Nonetheless, detailedhealing ki-neticsofa materialcanbecoupledtothepresentmodelthrough fracture properties andcrack filling behavior to simulate specific materials.

The paperisorganizedasfollows: theproposed crackhealing

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implementationaredescribedindetailinSection2.Veriﬁcationof the model is done through basic tests in Section 3, fromwhich salient features of the model are demonstrated.Section 4 is de-votedtotheapplicationofthemodeltoaparticle-basedextrinsic self-healing systemunder mechanical loading.Parametric studies are conductedtoshowcasetheapplicability ofthemodel consid-ering some realistic scenarios and the results are reported. Con-cludingremarksandfurtherworkarehighlightedinSection5. 2. Modelingoffractureandhealing

A cohesive zone-based fracture mechanics model is extended to model both the fracture and the healing in a uniﬁed consti-tutiverelation.Inthecurrentfracture-healingframework,the fol-lowingmodelingconsiderationsaremade.Firstly,thecrackhealing modeldoesnotincludehealingkineticsexplicitly,ratherthefocus istodevelopamethodologytosimulatecrackingandtherecovery ofmechanicalintegrityuponhealing. Consequently,wheneverthe healingprocessisactivatedatalocationwithinthecohesivecrack,

the resting period is assumed to be suﬃciently long such that

complete healingoccurs.This assumptionisnot necessaryper se, butenforcedinorderto havea specificfocusonrecovery of me-chanicalpropertiesusingamodifiedcohesiveconstitutiverelation. Nonetheless,dependinguponthetypeofhealingprocessinvolved ina specifichealingmaterial system, appropriatehealingkinetics canbetreatedseparatelyandcoupledwiththepresentframework. This,in turn,cangovern theeffectof parameterssuch asresting

time andtemperature on the degree of healing, which can then

befedasaninputtothepresentframeworkthroughappropriately deﬁnedfracturepropertiesofthehealedmaterialphases.This as-pectof healingkineticsis thesubjectofcompanion workby the authorsandtheircollaborators.

In thefollowingsubsection,the cohesive crackmodelwithout healingisdiscussedﬁrst,whichisthenfollowedbyadiscussionon thecohesivecrackhealingmodel.Thenumericalimplementationis addressedinthelastsub-section.

2.1. Cohesivecrackmodelwithouthealing

The cohesive zone model employed in thiswork corresponds

toa bilinearrelationgivenbyT,whichisascalarmeasure ofthe tractionttransmittedacrossthecohesivesurface,asafunctionof

,whichisascalarmeasureofthecohesivesurfaceopening dis-placement vector

δ

.Though severalother cohesive relationshave beenproposedintheliterature,abilinearrelationcapturesthe es-sentialingredientsofmostcohesiverelations,namelythecohesive strength

σ

c andthefractureenergyGc,whichareviewedas

vari-ablematerialpropertiesinthepresenthealingmodel.Thetraction

Tincreaseswithincreasingcohesivesurfaceopeningdisplacement

uptoamaximumvaluegivenby thestrength,

σ

c,and

eventu-ally decreases tozero, atwhich point thecohesive zone is fully-separatedinthesensethat no(positive)tractioncanbe transmit-tedacrossthesurface.Theinitial(increasing)partofthecohesive responseisusefulinconjunctionwithcohesiveelementsbutmay

be omitted for XFEM implementations where the cohesive

rela-tion is only activated when the critical value

σ

c is reached. The

areaunderthetraction–separationcurve,whichrepresentsthe to-talworkperunitareaexpendedincreatingafully-separatedcrack, correspondstothefractureenergyGc ofthematerial.

Aneffectivecrackopeningdisplacement variableisintroduced asfollows:

:=

δ

n

2+

γ

2

(

δ

s

)

2, (1)

where

δ

n and

δ

s are,respectively,thenormalandtangential

com-ponents ofthecrackopeningdisplacementvector

δ

resolved ina

coordinatesystemalignedwiththelocalnormalandtangential di-rections of a crack surface. In (1),

·

:=

(

· +

|

·

|

)

/2 refers to the Macaulaybracketand

γ

isanon-dimensionalweightingfactorfor thenormalandtangentialcontributionsgivenby

γ

:=

δ

n,0

δ

s,0 ,

where

δ

n,0and

δ

s,0denote,respectively,thecrackopeningatthe

onsetof failure fora pure normalanda pure tangentialopening withrespecttothecracksurface.

Inordertodeterminewhetherthecrackopeningisincreasing or decreasing due to the external loading process, the following loadingfunctionfisused:

f=_fˆ

₍

_,

_κ

₎

_:=

₋

_κ

_, ₍₂₎

where

κ

isa damagehistoryvariablethat,atagiventime t, cor-respondsto themaximumvalue attainedby theequivalentcrack openingduringaprocessuptothattime,i.e.,

κ

(

t

)

:=max

¯t∈[0,t]

¯t

.

The Karush–Kuhn–Tucker relationsfor theloading andunloading conditionscanbeexpressedasfollows:

f

κ

˙ =0, f≤ 0,

κ

˙ ≥ 0, (3) where

κ

˙ indicatesthe(time)rateofchangeofthedamagehistory variable with

κ

˙ >0 corresponding to an active damage step and

˙

κ

=0toan“elastic” step.

Theequivalentcrackopeningisusedtocomputetheequivalent tractionTas T=_Tˆ

₍

_,

_κ

₎

₌

⎧

⎨

⎩

ˆ g

()

if f =0and

κ

˙ >0, ˆ g

(κ

)

κ

otherwise, (4)

where gˆ is the effective traction–separation law. The upper and lowerexpressionsin(4)providetheequivalenttractionduring, re-spectively,crackgrowthandunloading/reloading.

Thespeciﬁc formofthe effectivetraction–separation lawused in the present work is a linear softening relation, which corre-spondsto

g=gˆ

()

=

σ

c

f−

i .

(5) Intheaboveexpression,theparameters

iand

f are,

respec-tively,theequivalentcrackopeningatthe onsetofsoftening and

themaximumequivalentcrackopening.Theseparametersmaybe

chosensuchthat, foragivenfracturestrength

σ

c,fracture

tough-nessGc andaninitialcohesivestiffnessK,

i=

σ

_Kc ,

f=

2Gc

σ

c ,

where the initially linearly “elastic” loading up to the fracture strengthin abi-linear lawcan bereproduced in(4) byassigning an initialdamage

κ

(

0

)

=

κ

0=

i.The parameters

i and

f are

chosensuchthatthemaximumofthefunctionTˆin(4)equalsthe fracturestrength

σ

c andthe integral ofTˆ from

= 0to

=

f

equalsthematerialfractureenergy(toughness)Gc.

After evaluating (4), the normal and shear tractions can be

computedas tn=

⎧

⎨

⎩

δ

n

T if

δ

n>0, K

δ

n if

δ

n<0, ts=

γ

2

δ

s

T, (6)

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2.2.Crackhealingmodel

The cohesive relation summarizedin theprevious section can beextendedtoformulateacohesivecrack-healingmodel.The sin-glehealingcaseisdiscussedﬁrst,whichisthenfollowedbya gen-eralizedmodelcapableofmultiplehealingevents.

2.2.1. Singlehealingevent

The proposed crack healing model is a composite-based

con-stitutive modelfor simulatingthe recovery of fracture properties upon activationof crackhealing. Thetraction components ofthe compositeresponse,t˜n andt˜s,areexpressedasaweightedsumof

thetractioncontributionsfromtheoriginalmaterial,t_n(0)andt_s(0),

andthehealingmaterialtn(1) andts(1),asfollows:

˜

tn=w(0)tn(0)+w(1)tn(1) t˜s=w(0)ts(0)+w(1)ts(1) (7)

wherethesuperscripts(0)and(1)representtheoriginaland heal-ingmaterials,respectively.Theweightingfactorsw(0)_and_w(1)

in-troducedin(7),which can take valuesbetween 0and1, are the

primary parameters in the model and can be interpreted asthe

surface-based volume fractionsof the original andhealing mate-rialrespectivelyattheinstanceofhealingactivation.Themeaning of“volume fraction” in thiscontext refers to the fraction ofthe crackareaoccupiedbyamaterialperunitcrackopening displace-ment.Ina two-dimensionalsetting,the crackarea fractionrefers tothecracklengthfractionperunitdepth.Asindicatedin(7),itis assumedthat apartiallydamagedarea thathasbeenhealed con-tainscontributionsfromboththeoriginalmaterialandthehealing material.Correspondingly,theweighting factorsw(0) _and_w(1) _are

relatedtofractionsofapartiallydamagedsurfacewherethe orig-inalmaterialis still capable oftransmitting aforce (inthe sense of a cohesive relation) while the healing material has occupied thecomplementary region.Observe that this assumption implies thatthe modelis essentially basedon an “equalstrain” distribu-tionamongthephases(Voigtmodel), inthiscasewiththe crack opening playing the role of a strain-like variable. Consequently, thetractions in each phasemay be overpredictedcompared toa modelbasedonan“equalstress” assumption(Reussmodel), how-everthecurrentVoigt-like modelpreserveskinematic compatibil-itywhereasaReuss-likemodeldoesnot.

In order to develop the constitutive model,deﬁne an energy-baseddamageparameterD(0)_as_follows:

D(0)

₍

_t

₎

_:₌ G(d0)

(

t

)

G(c0)

(8) whichrepresentsthe ratiobetweentheenergy dissipatedG(_d0)

(

t

)

duringdecohesion of the original material up to time t and the fractureenergyG(_c0)(workrequiredforcompletedecohesionofthe original material). Ina bilinear cohesive relation, aspresented in Section2.1,thisparametermaybeapproximatedas

D(0)

₍

_t

₎

_≈

κ

(0)

(

t

)

(0)

f

wheretheinitial,undamaged“elastic” responsehasbeenneglected (namelyitisassumedin(5)that

_f(0)

(0)

i ).

Prior tohealing, thecohesive response ischaracterized bythe cohesiverelationoftheoriginalmaterial,i.e.,withw(1)₌₀_in₍₇₎_.

Ifasinglehealingeventoccursatatimet=t∗,theproposed con-stitutivemodelassumesthatthefactorw(1) _is_given_by_the_value

oftheenergy-based damageparameteratthe instanceofhealing activation,D(0)∗_,_i.e., w(1)₌_D(0)∗ :=G (0)∗ d G(c0) . (9)

Correspondingly, a value w(1) ₌ ₀ _upon _healing _activation

repre-sentszeroequivalentdamaged area fractionofthe original mate-rialatagivenmaterialpoint(i.e.,theoriginalmaterialisfully in-tact) whilew(1) ₌ ₁_represents _a_{fully-damaged} _original _material

ata givenmaterialpoint (i.e.,thehealingmaterial wouldoccupy thefullydamagedmaterialpointuponhealing).Theinterpretation of(9)isthatthevolumefractionw(1)_available_for_the_healing

ma-terialinordertoﬁllandhealcanbedeterminedfromthevalueof the energy-based damage parameter ofthe original material, de-ﬁnedin(8)attheinstanceofthehealingactivation.Uponhealing of the available volume fraction w(1)_, _the _volume _fraction_of _the

original material, w(0)_, _assumes _a _value _equal _to ₁_{− w}(1)_, _which

isequaltotheequivalentundamagedareaoftheoriginalmaterial. Conversely,theenergy-baseddamageparametercanbeinterpreted asan equivalentdamaged area fraction ata givenmaterial point

in the context of the cohesive zone framework. A schematic of

thetraction–separation relations foramaterial pointunder dam-age andhealing is shown inFig. 2 depicting the features of the model.

Inaccordancewith(7)and(9),theeffectivefractureenergyG˜c

ofthecompositematerialafterhealingbecomestheweightedsum ofthefractureenergiesoftheoriginalandhealingmaterials,given as

˜

Gc=w(0)G(c0)+w(1)G(c1). (10)

2.2.2. Traction-crackopeningrelations:originalmaterial

The traction–separation relation corresponding to the

origi-nal material after healing is governed by a modiﬁed effective

displacement-basedcohesivecrackmodelexplainedasfollows:the effectivedisplacementfortheoriginalmaterialdefinedinthe con-ventionalcohesivezonemodelismodifiedbyintroducingshiftsin normalandtangentialcrackopeningdisplacementstotakeinto ac-count theeffectofhealing.These shiftsinthecrackopening dis-placementsleadtoamodifiedeffectivedisplacementforthe orig-inalmaterial,

(0)_,_given_as

(0)_:=

δ

n−

δ

(0)*n

2 +

γ

2

(

δ

s−

δ

s(0)*

)

2, t≥ t∗. (11)

Thereasonforintroducingtheshiftisasfollows:onactivation ofa healingprocess, thehealingagentdiffuses/ﬂowsthroughthe crackandcrack ﬁllingoccursthereby (fullyor partially)reducing thecrackopening.Asaresult,thecrackopeningdisplacements af-tercompletehealingshouldbeconsiderednominallyzero.To sim-ulatethisprocess,displacementshiftsareintroducedintothecrack openingdisplacements,whichmakethenominalopening displace-ment zeroupon completehealing. Further, a shiftisalso applied inthe crackopeninghistoryvariable,

κ

, whichisreset toits ini-tialvalue.Thisisdonetosimulatetheintactportionofthe origi-nalmaterialpoint,whereasthedamagedportionoftheconsidered materialpointisassumedtobehealedbythehealingmaterial.

In a cohesive-zonemodel, a partially-damaged material has a non-zerocrackopeningdisplacementbutitmaystillbecapableof transmittinga force. In the presentmodel,ifhealingis activated inapartiallydamaged surface,it isassumedthat theprocess oc-cursatconstantstressprovidedthereisnochangeintheexternal loading.

The shifts introduced in the normal and the tangential crack opening displacements forthe original material are givenas fol-lows:

δ

(0)∗ n =

δ

n∗− tn∗/

(

w(0)K

)

,

δ

(0)∗ s =

δ

s∗− ts∗/

(

w(0)K

)

. (12) Intheaboveexpressions,

δ

∗_n and

δ

∗_s aretheactual crack open-ing displacements inthe original material atthe instant ofcrack

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Fig. 2. Traction–separation laws of original and healing material, which upon weighted addition, results in a composite cohesive relation for the crack-healing model. healingactivation.Asshiftsincrackopeningdisplacementsare

in-troducedalongwithrestoration ofthecrackopeninghistory vari-able

κ

,theshiftsincrackopeningdisplacementsareconstructedin suchawaythatthetractions acrossthecohesivesurfacemaintain their continuity. Consequently,the traction componentst_n∗ andt_s∗

across thepartiallydamaged surfaceremainthesamebefore and afterhealingactivation.

The normal and shear tractions corresponding to the

origi-nal material during subsequent loading after healing are then

obtained from the corresponding traction–separation relations

(4)and(6)using theaforementioned equivalent opening

(0)_,_i.e.,

tn(0)=

⎧

⎪

⎨

⎪

⎩

(

δ

n−

δ

n(0)*

)

(0) T (0) _if

_δ

n>

δ

n(0)∗, K

(

δ

n−

δ

n(0)*

)

if

δ

n<

δ

n(0)∗, ts(0)=

γ

2

(

δ

s−

δ

(0)* s

)

(0) T(0). (13)

2.2.3. Traction-crackopeningrelations:healingmaterial

Similar to the original material, the traction–separation

rela-tions corresponding to the healing material are governed by a

modiﬁedequivalentdisplacementvariable,

(1)_deﬁned_as

(1)_:=

δ

n−

δ

n(1)*

2 +

γ

2

(

δ

_s₋

δ

(1)* s

)

2, (14)

where

δ

(1)*_n and

δ

_s(1)*areshiftsappliedtothetraction–separation relationofthehealingmaterial.Theshiftsareintroducedintothe

crackopeningdisplacementsofthehealingmaterialfollowingthe sameapproachasfortheoriginalmaterial.Themaindifferenceis thatthe healingmaterialisassumedtotransmit zeroloadatthe instantofhealingactivation.Thus,theshiftsincrackopening dis-placements forthe healingmaterial are theactual crack opening displacementsattheinstantofhealingactivation,i.e.,

δ

(1)∗ n =

δ

n∗

δ

(1)∗

s =

δ

s∗.

(15) Similar to the approach adopted for the original material after

healing, the normal and tangential traction components

corre-sponding to the healing material are obtained from an

equiva-lenttractionT(1)_of_the_{corresponding}_{traction–separation}_relations

(4)and(6)usingtheequivalentopening

(1)_,_i.e.,

tn(1)=

⎧

⎪

⎨

⎪

⎩

(

δ

n−

δ

(1)*n

)

(1) T(1) if

δ

n>

δ

n(1)∗, K

(δ

n−

δ

(n1)∗

)

if

δ

n<

δ

n(1)∗, ts(1)=

γ

2

(

δ

s−

δ

s(1)∗

)

(1) T(1). (16)

It is worth noticing that, in both the original andhealing mate-rialphases, theshifts inthecrackopeningdisplacements are ap-pliedatthecomponentlevel,i.e., individuallyon thenormaland tangentialcomponents.Thecompositetractionst˜nandt˜s,givenin

(7),areobtainedthrougharule-of-mixturesapproachanalogousto an equal strain assumption used forcomposite materials (in this caseanequalcrackopeningassumption)withmaterial-speciﬁc re-sponses givenby (13) and(16).This approach provides suﬃcient

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ﬂexibilityto specifyseparate materialpropertiesandfracture be-haviorfortheoriginalandhealingmaterials.

2.2.4. Multiplehealingevents

The approach presented in the previous section can be

ex-tendedtoaccount formultiple healingevents.Thisgeneralization is capable of dealing with a complex history of (partial) crack-ingsandhealings.Inthesequel,theindexpreferstothenumber ofhealingevents,rangingfrom0 tom,withtheconvention that

p=0 representsthe undamaged original state. The indexp may alsobeusedtorepresentthehealingmaterialphasethatisformed duringthepthhealingevent,againwiththeconventionthatp₌0 correspondstotheoriginalmaterial.Attheendofthemthhealing event,thecomposite-liketractioncomponentst˜n[m] andt˜s[m] ofthe

multiply-healedmaterialaregivenby ˜ tn[m]= m p=0 w[m](p)_t(p) n t˜s[m]= m p=0 w[m](p)_t(p) s (17)

wheretn(p)andts(p)arethenormalandtangentialtraction

compo-nentsofthepthmaterialphaseandw[m](p) _is_the_volume_fraction

ofthepthmaterialphase(indexinparentheses)presentorcreated atthemthhealing event(indexin square brackets). The relation givenin(17)isageneralizationof(7)forthecasem>1.For mod-elingpurposes, a separate indexis assignedto each newhealing materialcreatedat thepthhealingeventeven though theactual materials(chemicalcomposition)maybephysicallythesame.The purposeistokeeptrackoftheirindividualevolutionsthroughouta complexloading andhealingprocessstartingatpossiblydifferent states(i.e.,everyhealinginstanceisrecordedseparately).In accor-dancewiththeproposedconstitutivemodelforthesinglehealing event,itisassumedthatthevolumefractionw[m](p)_of_the_p_th

ma-terialphase isrelated to the energy-based damage parameter of thatphasepriortothemthhealingevent,whichcanbeexpressed recursivelyas w[m](p)₌

⎧

⎪

⎨

⎪

⎩

1 forp=m=0, w[m−1](p)

₍

₁_{− D}[m](p)

₎

_for₁_{≤ p}_<_m_, m−1 p=0 w[m−1](p)_D[m](p) _for₁_{≤ p}₌_m_. (18)

Intheaboveexpression,D[m](p) _is_the_value_of_energy-based

dam-ageparameterD(p) _{corresponding}_to _the_p_th_healing_phase _at_the

mth healingevent. With this notation, the term D(0)∗

in (9)can alternatively be expressed as D[1](0) _to _indicate _the _value _of _the

energy-baseddamageparameterevaluatedattheinstantatwhich theﬁrsthealingeventisactivated.It isalsotobe notedthat the sumofallw[m](p) _is_equal_to_1,_where_p_ranges_from₀_to_m_.

The fracture energy G˜[_cm] of a multiply-healed composite-like crack, which is a generalization of (10) for m>1, corresponds to the weighted sumof the fracture energies of the phases p= 0_,_._._._,m_{− 1}availablebeforehealingactivationandthefracture en-ergyofthelatestformedhealingmaterial p=m,i.e.,

˜ G[cm]= m p=0 w[m](p)G(cp). (19)

At the mth healing event, there are m+1 material phases at a material point within the cohesive zone for which the tractions in each phase are governed by the corresponding cohesive rela-tions.The shiftsinthe crackopeningdisplacements are obtained foreach phase such that thecontinuity ofthe tractions is main-tained within each phase, similar to the equations forthe shifts givenby(12)and(15).

For subsequent use, the expressions for the volume fractions

w[m](p) _in _the _case _of _two _healing _events _are _obtained _explicitly

from(18)withm=2,i.e.,

w[2](0)₌_w[1](0)

₍

₁_{− D}[2](0)

₎

₌

₍

₁_{− D}[1](0)

₎₍

₁_{− D}[2](0)

₎

_, w[2](1)₌_w[1](1)

₍

₁_{− D}[2](1)

₎

₌_D[1](0)

₍

₁_{− D}[2](1)

₎

_, w[2](2)₌_w[1](0)_D[2](0)₊_w[1](1)_D[2](1)

=

(

1− D[1](0)

₎

_D[2](0)₊_D[1](0)_D[2](1) _. ₍₂₀₎

The damagein theoriginal materialup to theﬁrsthealingevent isreﬂectedinthevalueD[1](0) _whereas_the_subsequent_damage_in

theoriginalmaterialandtheﬁrsthealingmaterialisaccountedfor, respectively,throughthevaluesD[2](0) _and_D[2](1)_._The

correspond-ingcompositetractionandfractureenergyafterthesecondhealing eventcanbecomputedfrom(17)and(19).

2.3. Numericalimplementation

The cohesivecrack healingmodeldescribedabove can be im-plementedinaﬁniteelementframeworkusingcohesiveelements (or using an XFEM approach). In the context of a ﬁnite element

solution procedure performed using a Newton–Raphson iterative

approach,thecontributionofthe cohesiveelementstotheglobal stiffnessmatrixisprovided bytheir element-wiseconsistent tan-gent matrix,which corresponds to the derivative of the traction vector with respect to the crack openingdisplacement. With re-spect to a local coordinate systemnormal (n) and tangential (s) tothecrack,thecomponentsofthetangentmatrixare,inviewof (17),givenbytheweightedconstitutive(material)tangentsofeach phasep,i.e.,

∂

t˜_i[m]

∂δ

j = m p=0 w[m](p)

∂

t (p) i

∂δ

j , i=n,s, j=n,s. (21) Correspondingly, the tangent matrix ofthe composite-like model requirestheindividual contributionsfromthephases.The expres-sionsfortheconstitutivestiffnesstangentsdependontheloading– unloading conditions, as indicated in (3), applied separately for eachphasep.

Undersofteningcondition:

For f(p)₌₀ _and

_κ

_˙(p)_>₀_, _the _components _of _the _consistent

tangentmatrixforthephasepareobtained,assumingthat

_i(p)

(p)

f ,from(4), (5), (11)and(13), as follows:

∂

tn(p)

∂δ

n =

σ

(p) c

1

(p)− 1

(p) f −

δ

n−

δ

n(p)∗

2

(

(p)

₎

3

,

∂

ts(p)

∂δ

s =

γ

2

_σ

(p) c

1

(p)− 1

(p) f −

γ

2

δ

s−

δ

s(p)∗

2

(

(p)

)

3

,

∂

tn(p)

∂δ

s =

∂

ts(p)

∂δ

n =−

γ

2

σ

(p) c

δ

n−

δ

n(p)∗

δ

s−

δ

s(p)∗

(

(p)

)

3 .

Underunloading/reloadingconditions:

Forf(p)_<₀_and

κ

_˙(p)₌₀_,_the_components_of_the_consistent

tan-gentmatrixare,assumingthat

(_ip)

(p)

f ,givenas

∂

tn(p)

∂δ

n =

σ

(p) c

1

κ

(p) − 1

(p) f

,

∂

ts(p)

∂δ

s =

γ

2

_σ

(p) c

1

κ

(p)− 1

(p) f

,

∂

tn(p)

∂δ

s =

∂

ts(p)

∂δ

n = 0.

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Fig. 3. Three element model: an initially zero thickness cohesive element placed in between two continuum two dimensional plane strain elements.

Thecaseoftheinitial“elastic” behaviorcaninprinciplebetreated formallyasareloadingcasebyassigningan initialdamage

κ

₀(p)=

(p)

i .However,sinceinthepreviousformulastheterm

(p)

i has

beenneglected,thetangentmatrixcanbeseparatelyspeciﬁedas

∂

tn(p)

∂δ

n = 1

γ

2

∂

ts(p)

∂δ

s = K ,

∂

tn(p)

∂δ

s =

∂

ts(p)

∂δ

n = 0,

where Kis thecohesive stiffness, assumedin thiscasetobe the sameforallphases.

3. Modelevaluationandtesting

In this section, the model is tested using a simple three-elementmeshasshowninFig.3,inwhichasinglezero-thickness

cohesive element is placed between two continuum elementsof

size20

μ

m _{× 20}

μ

meach.Severalaspectsareconsideredtostudy thebehavior ofthemodel,whichincludedifferentloading condi-tions,numberofhealingevents,propertiesofthehealingmaterial andthedegreeofdamage.

3.1. Fractureandhealinginmonotonicstraining

Intheﬁrstsetoftest simulations,thethree-elementsystemis subjected to various monotonic straining conditions to studythe behavior ofthecohesive elementunderdamageandhealing. The loading conditions are prescribed through applied displacements asshowninFig.3.Thetimehistoryofthespeciﬁeddisplacement is showninFig.4a,which correspondsto athree-stage deforma-tion,namely(i) anominallymode-Iopeningstageinvolvinga lin-ear increase inapplied deformation, (ii) a rest period with

heal-ing at the maximum deformation reachedin the ﬁrst stage and

(iii) a resumption of the mode-I deformation. In all the simula-tions,itisassumedthatcompletehealingoccursattheendofthe zero loading-rate time period (rest period). The post-healing be-haviorofthecohesiveelement correspondstothethirdstage.For the presentmodel, theactual duration of the restperiod hasno directeffecton thesimulations sincethehealingkineticsare not modeledexplicitly;onlythestateofthehealingmaterialupon re-sumption ofthe loading isrelevant. However, the healingperiod is indicatedin thetime historyforclarity andtoemphasize that healing is, in general, a process with time scales comparable to (andsometimeslargerthan)thetimescalesassociatedto mechan-icalloading.

Fortheconditionsassumedinthesimulation,theevolutionof thestressinthecohesivezoneinathree-elementsystemonly de-pendson thefracturepropertiesbutnotontheelasticproperties

of the bulk material. The fracture properties of the original and healingmaterialsaregivenas

σ

(0)

c =100MPa,

σ

c(1)=75,100MPa,

(f0)=

(1)

f =2

μ

m.

Asindicated above,two valuesareconsidered regardingthe frac-turestrength

σ

_c(1)ofthehealingmaterial,namelyalowerstrength comparedtotheoriginalmaterial (chosenas75% of

σ

_c(0)) andan equalstrength(100%). Thehealingmaterialsandtheoriginal ma-terialhavethesameﬁnalcrackopening

_f.Inprinciple,itis pos-sibletoconsider thecaseofa healingmaterialthat hasa higher strength,butinordertostudythatsituation,itismorerelevantto carryoutananalysiswithmorethanonecohesiveelementwhere asecondarycrackisallowedtoinitiateelsewhere.

Theeffectivetractionasafunction oftime isshowninFig.4b for both values of the fracture strength of the healing material. As showninthe ﬁgure,the traction inthe cohesive element ini-tially increases up to the value equal to cohesive strength

σ

c(0)

of the original material and then decreases as a result of

dam-age evolution. Complete failure occurs when the crack opening

reachesthe critical crackopening forfailure

(_f0) ofthe original phase.Afterwards,thecrackopeningcontinuestogrowduetothe externally-imposeddeformationatessentiallyzerostress.As indi-catedinFig.4a,healingisactivatedatt=100s (andisassumed tobecompletedatt=300s).Correspondingly,ashiftinthecrack opening of

δ

n(1)∗=3

μ

m and

δ

s(1)∗=0 is taken into account and

theloadingisresumed.Themaximumloadisreachedatthe corre-spondingvalueofthefracturestrengthofthehealingmaterial

σ

c(1)

forbothcasesconsidered(75%and100%).Theloadingcontinuesin thesoftening(degradation)regimeuntilthehealingmaterial com-pletelyfails,whichoccurswhentheshiftedcrackopeningissuch that

(1)₌

(1)

f .

Theeffectofthedamageduetoloading andtherecovery due tohealingcanbeseeninFig.4cintermsoftherelevantdamage variablesasafunctionoftime.Duringtheﬁrststageofloading,the damagevariableD(0)_of_the_original_material_increases_from₀_to_1,

whichindicates that theoriginal materialundergoes full damage. Thehealingprocess inthesecond stage isnotmodeled explicitly butratherprovidedasinputforthecohesivemodeltoanalyzethe recoveryofstrength.Duringthatprocess,thedamagevariableD(0)

of the original material remains at 1 while the damage variable

D(1) _of _the _healing _material _becomes_active _with _an _initial _value

equaltozero (nodamage). Duringthe thirdstage ofthe process, thedamagevariableD(1)_of_the_healing_material_increases_from₀

to1,henceattheendoftheloadingprocessthehealingmaterial isalsofullydamaged.

Theresultsofthe simulationcan alsobereported intermsof thetractionacrossthecohesiveinterfaceasafunctionofthecrack openingdisplacement as shown inFig. 4d. Although the loading process inthis exampleis relatively simple,it illustrates the im-portanceofusingtheshiftinthecrackopeningvariableto prop-erlysimulatetheevolutionofstressduringhealing.Indeed,asmay beobservedinFig.4d,thematerialfollowstheexpectedcohesive responsestartingfromthevalue

(1)₌₃

_μ

_m_as_a_new_origin_after

healing.

3.2.Multiplehealingofapartially-damagedmaterial

The next example to illustrate the features of themodel per-tainsto multiple healing ofa partially-damaged material. In this case, the material is loaded andhealed according to the applied deformationshowninFig.5a. Asindicated intheﬁgure,the ma-terial is initially extended and undergoes partial damage. Subse-quently,thematerialishealedandtheloadingisresumed,which generates partial damage of the original and the healing mate-rial.The materialthen experiencesa second healingeventbefore

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Fig. 4. Case 1: Illustration of response of cohesive element under monotonic straining, healing and further straining. The response includes the cases of healing materials with fracture strengths equal 75% and 100% of the strength of the original material.

loadingisresumed untilﬁnal failure.In thissimulation, the frac-turestrengthofthehealingmaterialsarechosen as

σ

c(1)=

σ

c(2)=

0.75

σ

c(0) withtheactual valuesasindicated intheprevious

sub-section.

The evolutionof theeffective(composite)normaltractiont˜n[m]

andthephase-weightedtractions w[m](p)_t(p)

n areshowninFig.5b

asafunctionoftime.Theeffectofhealingonthefracturestrength isreﬂected inthe distinct peak values ofthe response. The ﬁrst peakcorrespondstothefracturestrength

σ

c(0)oftheoriginal

ma-terial.The second peak liesbetween theone of theoriginal ma-terialandthe healedmaterialsincethe originalmaterialwasnot fullydamaged before healingwas activated,hence it partly con-tributestotheeffective(composite)fracturestrength.Inthatcase thestrength is w[1](0)_t(0)

n +w[1](1)

σ

c(1), withtn(0) beingthe stress

ontheoriginal materialatthe instantthat thestress ontheﬁrst healingmaterialreachesitscriticalvalue.Similarly,thethirdpeak containscontributionsfromthethreephasesthatareactiveatthat instant,namelyit isgivenby w[2](0)_t(0)

n +w[2](1)tn(1)+w[2](2)

σ

c(2),

withtn(0)andtn(1)beingthestressesontheoriginalandﬁrst

heal-ing material atthe instant that the stress on the second healing materialreachesitscriticalvalue.

Theevolutionofthedamageparameterforeachphaseisshown inFig.5c,whereitcanbeseenthattherateofdamageperphase decreasesasthenumberofactivephasesincreases(i.e.,the dam-agegets distributed amongthe differentphases, withthe largest ratecorrespondingtothemostrecentlycreatedphase).The effec-tivetraction–separationrelationformultiplehealingofa partially-damagedmaterialisshowninFig.5d.Theeffectivefractureenergy, asgivenby(19),dependsonthenumberofhealingeventsandthe volumefractions w[m](p)_._As _may_be_inferred _from_Fig.₅_d,_the

to-talfractureenergyofthematerial,measuredastheareaunderthe curve,increasesasaresultofthehealingprocesscomparedtothe originalmaterial.

3.3. Unloadingafterhealingofpartially-damagedmaterial

In order to validate the unloading features of the cohesive

model, the three-element system is subjected to a loading and

healing process as indicated in Fig. 6a. In this case the material is extended, undergoes partial damage followed by healing, after whichextension isresumed andﬁnally thematerial is unloaded. The unloading is speciﬁed in terms ofdisplacement, which ends at the displacement for which the stress vanishes. The fracture

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Fig. 5. Case 2: Illustration of response of cohesive element considering partial damage and healing. The fracture strength of the healing material is assumed to be equal to 75% of that of the original material.

Fig. 6. Case 3: Partial damage, healing, partial damage and unloading. The fracture strength of the healing material is assumed to be equal to 75% of that of the original material.

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Fig. 7. Geometry and ﬁnite element model of a unit cell of an extrinsic self-healing material. The unit cell is subjected to a nominal mode I loading. A small precrack is used to guide a matrix crack towards the particle. A layer of cohesive elements is placed to allow for crack propagation in a predeﬁned direction given by the initial precrack.

strengthofthehealingmaterialistakenas

σ

c(1)=0.75

σ

c(0).The

ef-fectivetractionasafunctionofcrackopeningisshowninFig.6b. Asmaybe observed inthe ﬁgure,upon unloading, the stress re-turnselastically tozero alongthe corresponding unloading curve

of each phase, which depends on the damage parameter of the

speciﬁcphase.Thestressiszeroatthecrackopeningdisplacement atwhichhealingoccurredaccordingtotheshiftparametersgiven in(12)and(15).

Other loading cases, not shownhere for conciseness,indicate thatthemodelisable topredictthebehavior undercomplex

se-quencesof mixed-mode loading, unloading andhealing. The

ap-plication ofthe healingcohesive modelunder non-homogeneous

conditionsisanalyzedinthefollowingsection.

4. Applicationtoparticle-basedextrinsicself-healingmaterial Inthissection,thecohesivehealingmodelisappliedtoaunit cell of an extrinsic self-healing material in which a single

heal-ing particle is embedded within a matrix material as shown in

Fig.7. In extrinsicsystems, the particle containsa healing agent (i.e., the material contained inside the particle) that is normally protectedby an encapsulation systemto prevent premature acti-vation of the healing process. The working principle of this sys-temisthatthehealingmechanismisactivatedwhenacrackthat propagatesthrough thematrixinteractswiththeparticle,usually breakingthe encapsulationand allowing transport ofthe healing

agent through the crack. Some self-healing system may involve

auxiliarymaterialsthat arenecessaryfortriggeringand/or partic-ipatinginasubsequentchemical reactiontocreatethefinal form ofthehealingmaterial. Thepresentsimulation assumesthat any additionalsubstancerequiredfortheprocessisreadilyavailablein thematrixmaterial(e.g.,free oxygentransportedbydiffusion re-quiredforoxidationasfoundinself-healingthermalbarrier coat-ingsSloofetal., 2015).Distinct cohesiverelations can beused at different spatiallocations (matrix, particle, matrix–particle inter-face),hencephase-specific fracturepropertiescanbespecifiedfor thehealingagentinsidetheparticleandthehealingmaterialthat appearsinthecracksafteractivationofthehealingmechanism.

As shown in Fig. 7, the unit cell used in the simulations is anL× L domainwitha circularparticleofa diameterd=2r.For

simplicitya two-dimensional computational domain under plane

strainconditionsischosen,meaningthattheparticleshouldbe in-terpretedasprismatic(ﬁberintheout-of-planedirection).Despite thisinterpretation,themodelisassumedtobequalitatively repre-sentativeofasphericalparticlealbeitwithadifferentvolume frac-tion.Inthesimulations,thelengthischosenasL=75

μ

mandthe

diameterasd=10

μ

m,whichcorrespondstoanominal(in-plane) particlevolumefractionof14%.Inthefiniteelementmesh, cohe-sive elements are inserted along a horizontal plane in the mid-height ofthe model,by which the crackis allowed to propagate alongthepre-definedpath.Inprinciple,arbitrarycrackgrowthcan bemodeledbyinsertingcohesiveelementsalongallbulkelements inmesh,althoughthatapproachisnotrequiredforpurposesofthe presentstudy(Ponnusami etal., 2015b). Thefiniteelement mesh issufficientlyresolvedsothatproperdiscretizationofthecohesive zone isensured.Displacement-drivennominalmode Iloadis ap-pliedby specifyingvertical displacementsatthe cornernodeson therightsideofthedomainwhilethecornernodesattheleftare fixedasshowninthefigure.Boththematrixandthehealing par-ticleareassumedtobeisotropicandlinearlyelasticuptofracture. Forthesakeofsimplicity,thematerialproperties(bothelasticand fracture)ofthematrixandthehealingparticlearekeptthesame andthevaluesaregivenasfollows:

Em₌_Ep₌₁₅₀_GPa_,

_ν

m₌

_ν

p₌₀_.₂₅_,

σ

m c =

σ

p c =

σ

c(0)=400MPa, Gmc =G p c =G(c0)=100J/m2,

where E and

ν

refer to Young’s modulus and Poisson’s ratio, re-spectively, andthe superscripts mand p referto the matrixand the particle,respectively. Since theproperties ofthe particleand thematrixareassumedtobeequal,theoriginalmaterial,as indi-catedbythesuperscript0,referstoeithertheparticleorthe ma-trixdependingonlocation.Theinterfacebetweentheparticleand the matrixisassumed tobe perfectlybondedandinterface frac-tureistakennottooccur.It isworth pointingoutthat ingeneral theelastic andfracturepropertiesofthe healingparticleandthe matrixare different,which infactdecide whetheramatrixcrack would break the healingparticle or not. This aspect of a matrix crackinteractingwithhealingparticlesofdifferentproperties com-paredtothe matrixisdealtindetailinPonnusami etal.(2015b) but isnot relevant forthe simulations presented in this section. Instead,emphasisisplacedhereonhowthecrackhealing behav-ior affects the recovery of mechanical properties of the material system. Further,asindicated above,thefracture propertiesofthe healingparticleinitsinitialstateareingeneraldifferentthanthe propertiesof thehealing materialthat ﬁlls thecracks, which are speciﬁedseparatelyasexplainedinthesequel.

Several parametric studies are conducted to evaluate the be-havior of the unit cell and the results in terms of global load-displacement response are reported inthe following subsections. Intheﬁrstsubsection,simulationsareconductedtostudythe ef-fect of variations in the fracture properties ofthe healing mate-rial.Inthesecond subsection,aparametricstudyisperformedto

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Fig. 8. Healing under unloaded condition: applied loading to unit cell and reaction force as a function of applied displacement for various values of the fracture properties of the healed material.

understandhowdoestheavailableamountofhealingagentaffect thecrackhealingbehavior. Inthethirdsubsection,multiple heal-ing events are simulatedand theresulting load-displacement re-sponseisreported.

4.1. Effectofpropertiesofhealingmaterialandhealingconditions

The fracture propertiesof thehealing material,formed asthe resultofthehealingprocess,areoftendifferentfromthe surround-ing host material.The fracture propertiesof thehealed zone de-pend on thetime available forhealingand theproperties ofthe healing product.A second aspect that is relevantfor the healing processistheloadingconditionsduringhealing. Healingisa pro-cessthattypicallyrequirestimetooccur,andtheefficiencyofthe process is often connected to providing a sufficiently long “rest time” inwhichtheloadingrateiszeroandchemicalreactionshave sufficienttimetobecompleted.However,evenifasufficient“rest time” is provided,the(constant)loadingstateinfluences the sub-sequentmaterial response ofthehealed material.In thissection, two representative loading states during healing are considered, namelyhealingunderzero-stress(unloaded)conditionsand heal-ing underfixed applied displacement(constant loadduring heal-ing). Different properties for the healingmaterial are considered foreachloadingstateduringhealing.

4.1.1. Healingunderunloadedcondition

Intheliterature,mostexperimentalstudiesdealwithtest pro-tocols in whichthe sampleisunloadedandallowed to returnto its unstrained state, hence healingoccurs underunloaded condi-tions (Kessler et al., 2003; Brown et al., 2002; Williams et al., 2007;Pangand Bond, 2005b; Songet al., 2008). In orderto an-alyze the predictions of the model under similar conditions, the unit cellshowninFig.7issubjectedtoaloadingandhealing se-quence asindicated inFig. 8a. Under this loading, the specimen ispartiallyfracturedandthenunloaded.Healingisallowedto oc-curin theunloadedcondition, whichis thenfollowed by reload-ingofthehealedspecimen.Theresponseoftheunitcellinterms of theapplied vertical displacement andthe corresponding reac-tionforceisshowninFig.8bforvariousfracturepropertiesofthe healingmaterial,namely

σ

_c(1)/

σ

(0)

c ,G(c1)/Gc(0)=0.25,0.5,0.75and

1,wherethesuperscript1referstothehealingmaterial.Asshown inFig.8b,thecurvecorrespondingtoequalpropertiesofthe heal-ing and original material predicts a recovery of the response af-ter healingsimilar tothatoftheoriginal material.Thenextthree

curvescorrespondtolowervaluesofthefracturepropertiesofthe healingmaterial andhencethe load-displacementcurves fall be-lowthatoftheoriginalmaterialafterhealing.

Itistobementionedthat,afterhealing,recrackingoccursalong thesamepathastheinitialcrack.Thisisduetothefactthe frac-turepropertiesofthe healingmaterialare lower than oratleast equal to that of the original material properties. Nonetheless, if thepropertiesofthehealingmaterialare higherthanthat ofthe originalmaterial,thecrackwouldpropagatealongadifferentpath

which is weaker than the healed zone. However, the recovered

load-displacementresponse wouldbe similartothe onewiththe samefractureproperties,asthecrackistraversingalongthe origi-nalmaterial.

4.1.2. Healingunderconstantloadcondition

In situations of practical interest, healing may occur under a non-zero load, which impliesthat thecrack opening is non-zero asthe healingmaterial ﬁlls thecrack gap.To studythe effectof the loading state duringhealing on the post-healing response of the material, simulations are carried out according to the load-ingsequenceshowninFig.9a.In thiscase, thespecimenis (par-tially) fractured, allowed to heal at a constant applied displace-mentandsubsequentlyreloaded.As inthepreviouscase(healing at unloaded conditions), four different fracture properties of the healingmaterialareconsidered,givenbythestrengthandfracture energyratios

σ

_c(1)_/

σ

_c(0)_,G_c(1)_/G(_c0)₌0.25,0.5,0.75and1.Theload carrying capability of the healed specimens is shown in Fig. 9b, whichindicatesthereactionforce oftheunit cellasafunctionof theappliedverticaldisplacement.

Foreach setofmaterial propertiesofthehealingmaterial,the stateofthespecimenisthesamepriortohealing.Afterhealingat a constant crackopening proﬁle, thespecimen recovers its load-carryingcapabilityasshowninFig.9a.Itcanbeobservedthatthe post-healingforcepeakishigherthantheforcepeakofthe origi-nalmaterialforthecasewhenthehealingmaterialhasthesame fracturepropertiesoftheoriginalmaterial.Thisresultispartlydue totheequalstrainkinematicassumptionoftheVoigt-like compos-itemodel,asindicatedinSection2.2.1,whichtendstooverpredict theforceresponse.Itisanticipatedthata morecomplex

compos-itemodel,whichpreservesbothlinearmomentumandkinematic

compatibility,would predict a lower post-healing peak. Although thepresentmodelprovidesan upperestimate ofthepost-healed behavior,itallowstocomparetheeffectofthestateofthe mate-rialduringhealingonthepost-healingbehavior. Inparticular,the

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Fig. 9. Healing under constant loading condition: applied loading to unit cell and reaction force as a function of applied displacement for various values of the fracture properties of the healed material.

Fig. 10. Unit cell specimen at the ﬁnal state of the applied loading given in Fig. 9 a. Representative local response curves illustrate how the introduction of a shift in the (local) crack opening displacement accounts for the proper origin upon resumption of the load after healing.

post-healingfailureinthecaseofhealingunderloadedconditions ismore sudden (i.e., qualitatively more brittle,see Fig. 9b) com-paredtothecaseofhealingatanunloadedstate(seeFig.9b), ex-ceptwhenthepropertiesofthehealingmaterialarerelativelylow. To gain more insight in the healing process under constant crackopening proﬁle, thelocal response curves atselected loca-tions are shown in Fig. 10 for the case when the properties of the healing material are

σ

c(1)/

σ

c(0) = Gc(1)/Gc(0)=0.5. As can be

observedinthe ﬁgure,the crackopenings attheinstant of heal-ing depend on location, with increasing values towards the side wheretheopeningloadisapplied.Observethat somepoints un-dergohealingfromafully-failedstate,somefromapartially-failed

state andsome pointsthat arecracked attheendof theloading processexperiencednohealingsincetheyhadnotfailedatthe in-stantatwhichhealingwasactivated.

Afterhealing,theeffectivecrackopeningdisplacementsbecome zerointhehealedzoneduetothedisplacementshiftsintroduced inthemodeltoaccountforcrackﬁlling.However,modelingofthe crack gap ﬁlling andhealing is only done implicitly through the shiftincrackopeningdisplacements,wherebythenewmaterialis notexplicitlymodeledasanadditionalmaterial(ormass)entering thesystem.

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Fig. 11. Effect of ﬁlling eﬃciency: applied loading to unit cell and reaction force as a function of applied displacement for various values of healed areas.

4.2. Degreeofcrackﬁllingandhealing

Intheprevioussubsection,itwasassumedthatthehealing par-ticle, uponfracture,releasesanamountofhealingagent(denoted as Vh) that issuﬃcient for complete ﬁllingof the crack opening

volume (orcrackopeningarea intwo dimension,denoted asVc).

However, dependingonthe amountofavailable healingmaterial, thegeometricalcharacteristicsofthecrackandthemodeof trans-port, it mayoccur that thecrack isonly partiallyﬁlled, whichis akeyfactoraffectingthehealingcharacteristicsandhencethe re-covery of mechanicalproperties.In thissection, theeffect ofthe ratioVh/Vc ofhealingagentavailabletotherequiredhealingagent

forcompletefillingisstudied.Here,thevolumeofrequiredhealing agentforcompletecrackfillingreferstothetotalcrackvolumeat the instanceofhealingactivation. Theratio consideredis generic andits interpretationinaspecificself-healingmaterialsystem re-quiresunderstandingofitshealingcharacteristics.Forinstance,the amount ofhealingmaterialproduced astheresultofthe healing process is directly relatednot onlyto the volume of the healing particle,butalso thereaction kineticsofthe healingprocess. For example, inone ofthe extrinsicself-healing systems reportedin theliterature,thehealingagentwithintheparticleproduces heal-ing material throughincrease involume by oxidationunder high temperatureconditions(Sloofetal.,2015;Ponnusamietal.,2015a). Hence,thetermVhmeansherethevolumeofthehealingproduct

formedastheresultofthehealingprocess,whichisusedfor heal-ing the crack.The notionofcomplete orpartial ﬁlling,measured by therationVh/Vc,refersto theamountofcrackgapﬁlled with

healingmaterialregardlessofthefracturepropertiesofthehealing material.

Simulations are conductedforfour differenthealingprocesses with ﬁlling eﬃciencies of V_h_/Vc= 1, 0.75, 0.5 and 0.25. In all

casesthepropertiesofthehealingmaterialaretakenas

σ

c(1)/

σ

c(0),

G(_c1)_/G_c(0) ₌ 0.75. The specimens are loaded according to the

sequence indicated in Fig. 11a and the corresponding force–

displacementcurvesareshowninFig.11b.Tointerprettheresults forpartialfilling,itisusefultoindicatethespatiallocationwhere healingoccurs,whichisshowninFig.12.Asindicated inthe fig-ure, filling is assumed to take place in the zone adjacent to the healingparticle.

The curve showninFig. 11b corresponding toa complete ﬁll-ing (ﬁlling ratioVh/Vc=1) representsthe highestpossible

recov-ery of the load-carrying capability for the given fracture proper-tiesofthehealingmaterial.Asexpected,therecoveryofthe

load-carryingcapabilitydecreases withdecreasingﬁllingratiosVh/Vc=

0.75,0.5,0.25.Onerelevantdifference betweentheeffectofa de-creaseinfillingefficiencyandadecreaseinthefractureproperties ofthe healingmaterial fora fixed fillingefficiency isthat in the lattercasetheinitialslopeofthepost-healedbehaviorremainsthe same fordistinct fracture properties whereas in the former case theinitialslope decreaseswithdecreasing fillingefficiency (com-pare Figs. 9b and11b). Inthe simulations showninFig. 11b,the initialpost-healingslopereflectstheincreaseincompliancedueto purelygeometrical effects.Theun-healed portionofthe crack fa-cilitatesthe(elastic)deformationofthespecimen.Thiseffectmay potentiallybeusedintheinterpretationofexperimentalcurvesas anindicationofpartialfillingofacrack.

4.3.Multiplehealingevents

Some materials with intrinsic self-healing capacity (such as

MAX phases), may undergo multiple healing events whereby a

crackishealedon multipleoccasions(Lietal.,2012; Sloofetal., 2016). In extrinsic systems, multiple healing may occur in cases wherethereisan externalsupplyofhealingmaterial,butalsoin particle-basedsystems when inactivated particles (or portions of partially activated particles) can still supply healing material for anadditionalhealingevent. Inthissection,itisassumedthat the particleintheunitcellshowninFig.7iscapableofproviding suf-ficient healing material for two healing events. The specimen is subjectedto a loading andhealing process asshownin Fig. 13a. Theratioofcrackopeningvolume (orarea)totheavailable heal-ing material volume isassumed to be 1for both healingevents, resulting incomplete filling of the crack. The fracture properties ofthe healing material after the first healingeventare takenas 75%ofthoseoftheoriginalmaterial,whileforthesecondhealing event,thepropertiesaretakenas50%ofthatoftheoriginal mate-rial,hence

σ

_c(1)_/

σ

_c(0)_,G(_c1)_/G(_c0)₌ 0.75and

σ

_c(2)_/

σ

_c(0)_,G(_c2)_/G(_c0)₌

0.5.Thisassumptionismeanttoimplicitlyrepresentadegradation onthequalityofthehealingmaterialafterthefirsthealingevent. The reaction force on the unit cell is shown in Fig. 13b as a functionoftheapplied verticaldisplacement.As canbe observed inthefigure, theloadcarryingcapacitymaybe (partially) recov-eredmultipletimesprovidedthattheself-healingmechanism sup-plies sufficient healing material for multiple healing events. Al-thoughasinglehealingeventcannaturallyextendthelifetimeof amaterial,amoresignificant extensioncanbe achievedina ma-terialcapableofmultipleself-healingrepairs,evenifthequalityof the healing material degrades during subsequent healing events.