Modelling fracture of aged graphite bricks under radiation and temperature

ContentslistsavailableatScienceDirect

Nuclear

Materials

and

Energy

journalhomepage:www.elsevier.com/locate/nme

Modelling

fracture

of

aged

graphite

bricks

under

radiation

and

temperature

Atheer

Hashim

a,b

_,

_Si

_Kyaw

a,c,∗

_,

_Wei

_Sun

a

a _{Department of Mechanical, Materials and Manufacturing, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK} b _{College of Water Resources Engineering, Al Qasim Green University, 8, Al Qasim, Iraq}

c _{Department of Engineering & Mathematics, Faculty of ACES, Sheffield Hallam University, Sheffield, S1 1WB, UK}

a

r

t

i

c

l

e

i

n

f

o

Article history: Received17August2016 Revised4March2017 Accepted7March2017 Availableonlinexxx

Keywords: Nucleargraphite Neutrondose Irradiationtemperature Damagemodel Crackgrowth

a

b

s

t

r

a

c

t

Thegraphite bricksoftheUKcarbondioxidegas coolednuclearreactorsaresubjectedtoneutron ir-radiationand radiolyticoxidationduringoperationwhichwillaffectthermal and mechanicalmaterial propertiesandmayleadtostructuralfailure.Inthispaper,anempiricalequationisobtainedandused torepresentthereductioninthethermalconductivityasaresultoftemperatureandneutrondose.A2D finiteelementthermalanalysiswascarriedoutusingAbaqustoobtaintemperaturedistributionacross thegraphitebrick.Althoughthermalconductivitycouldbereducedbyupto75%undercertain condi-tionsofdoseandtemperature,analysis hasshownthatithasnosignificanteffectonthetemperature distribution.Itwasfoundthatthetemperaturedistributionwithinthegraphitebrickisnon-radial, dif-ferentfromthesteadystatetemperaturedistributionusedinthepreviousstudies[1,2].Toinvestigate thesignificanceofthisnon-radialtemperaturedistributiononthefailureofgraphitebricks,asubsequent mechanicalanalysiswasalsocarriedoutwiththenodaltemperatureinformationobtainedfromthe ther-malanalysis.Topredicttheformationofcrackswithinthebrickandthesubsequentpropagation,a lin-eartraction–separationcohesivemodelinconjunctionwiththeextendedfiniteelementmethod(XFEM) isused.Comparedtotheanalysis withsteadystateradialtemperaturedistribution,thecrackinitiation timeforthemodel withnon-radialtemperaturedistributionisdelayed byalmostone yearinservice, andthemaximumcracklengthisalsoshorterbyaround20%.

© 2017TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Nowadays, inthe UK,the gas-cooled graphitemoderated type dominatesover90%ofthenuclearreactor[3].Graphiteisalso con-sideredtobeusedwithinthedesignofthenewreactorgeneration (suchas theHeliumgascooled very hightemperaturereactor or VHTR) as structuralsupport andasmoderator because of its ex-tremepurity,abilitytowithstandextremelyhightemperatures[4]. The corestructure ofthereactor isformed fromjoininggraphite bricks, with keys and keyways. During operation, graphite is ex-posed to neutronradiation andtemperature gradients that could cause irradiation and radiolytic oxidation.These occurrences will affectmaterialpropertiessuchasweightloss,porositychangesand thermal conductivity which can potentially lead to material and structuralfailure[5].

∗ _{Correspondingauthorat:DepartmentofEngineering&Mathematics,Facultyof} ACES,SheffieldHallamUniversity,Sheffield,S11WB,UK.

E-mail addresses: si.kyaw@hotmail.com,s.t.kyaw@shu.ac.uk(S.Kyaw).

FuturereactorssuchasVHTRareexpectedtooperateathigher temperatures than the current reactors and its core outlet tem-perature (COT) will be about 1000°C [5–8]. In orderto enhance thermalefficiencyoffuturenuclearreactors,extensiveresearchhas beencarriedout,aimingtoprovideanimproveddesignforfuture reactors[6–10].Asthereactorisexposedtoneutrondoseand tem-perature,manymaterialpropertiesarechangedbyirradiationand radiolytic oxidation ifairor carbon dioxideis used asa coolant. Some of the properties of nuclear graphite which change under service conditions include elastic properties and thermal expan-sioncoefficient,irradiation creep,irradiation induced dimensional changes. The prediction of the nuclear graphite lifetime and its structuralintegrityaresignificantissuesforthesafetyand reliabil-ityofreactoroperationthatcouldbeaffectedbychanging proper-ties.

Inpreviousstudies[2],aconstitutivemodelhasbeendeveloped forthe effects of mechanical properties changes on stress states within the graphite brick. Resultant crackdevelopment from the stressstate within thebrickwasalsostudied in[1].Forprevious studies, steadystate temperaturedistribution wasused to

calcu-http://dx.doi.org/10.1016/j.nme.2017.03.038

2352-1791/© 2017TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

latethermalstressassumingthatthermalconductivityofgraphite brick remains constant across the brick during the operation. It wasalsoassumedthatthetemperaturedistributionislinearacross thegraphitebrick.However,studies [11,12]haveshownthat ther-malconductivityofnuclear graphitechanges withboth tempera-tureandfastneutronirradiation.Hence,anewthermalanalysisis requiredtocalculatethetemperaturegradientacrossthebrickby consideringthe changesin thermalconductivitydueto doseand temperature.

Inthispaper,anempiricalequationfortheevolutionofthermal conductivityisderivedfromavailableexperimental databy Mars-denetal.[12].Usingthisrelationshipwithrelevantthermal prop-erties,a thermal analysis is carriedout. Resultantnodal temper-aturesare then implemented into mechanicalfinite element (FE) model with a constitutive model developed in [2] and fracture mechanicsbased damagemodel developedin [1].Stress analyses andcrackdevelopmentarethencarriedout.Comparisonsarealso made to models with the linear temperature distribution across thebrick.

2. Empiricalrelationshipofthermalconductivityofgraphite withneutrondoseandirradiatedtemperature

Theeffectsoftemperaturegradientsandirradiation caninduce damagetothegraphitebricksandthisleadstochangesinits ther-malconductivity.Thecommonapproachisbasedonrelating ther-malconductivity(K)tosummationofthermalresistances(1/K)as aresultofscatteringhurdles[13,14]

K

(

x,T

)

=

₁

α

(

x

)

KGB+

1 Ku+

1 KRD

(1)

wherethe term

α

(x) is a coefficient, which is function of orien-tation terms (basal plane, x), 1

K_GB is the thermal resistance due

to grain boundary scattering, 1

Ku is the thermal resistance due

toUmklappscatteringorthe so-calledphonon-phononscattering

[14],and _K1

RD is thethermalresistance dueto lattice defect

scat-tering.KGB dependson the graphite perfection andis taken into

accountonlyforlowertemperatures,asitisinsignificantat inter-mediateand high temperatures.Ku can be scaled asa quadratic

functionoftemperature [14]anditcontrols thermalconductivity athigher temperatures.Finally,KRD dependson theneutron

irra-diation,asthelatter plays a significantrole in producingvarious typesof defects,which are more effectivein scatteringphonons. Therefore, many studies [5,13,15] emphasise that the change in conductivityisonlycausedbythephononscatteringinlattice de-fects.Thisphonon scatteringincreaseswithtemperature.Henceit isnecessarytoconsiderboth irradiationdoseandtemperatureto estimatethechangeinthermalconductivity.

For significant nuclearirradiated graphite grades e.g. Gilsocar-bon,Kelly[11]proposedaformulaasshowninEq(2),tocalculate thethermalconductivity.It includesthe terms,fandS_k,obtained empiricallyasafunctionofdoseandtemperature[12].

1 K

(

T

)

=

1 Ko

(

30

)

Ko

(

30

)

Ko

(

T

)

+

f.

δ

(

T

)

Sk (2)

wherefisthe initialirradiation inducedchangeequal to Ko(30) Ki(30)= Ko(30)

K(30) −1 and Sk is a structural term which allows decrease in

thermal conductivity at higher dose. Marsden et al. [12] used

Eq(2)onirradiated andunirradiatedgraphite samples.The ther-malconductivity isshown to be directly proportional to the op-erationtemperature(T)forirradiated graphitebuttheopposite is trueforunirradiatedgraphite.

Without using an atomistic physical approach for calculating ofthe variation of thermal conductivityfor the graphite such as

Fig.1. Thermalresistivityvariationwithfastneutrondoseasafunctionofdose andtemperature[12].

the one proposed in [16], the thermal conductivity dependence ondoseandtemperaturecanbedeductedfromexperimentaldata curvesorincludeempiricalterms obtainedfromtheexperiments using Material Test Reactor (MTR). In the currentpaper, thermal conductivityvariationresultsofMTRfromMarsdenetal.[12]are used to generate the empirical equation. The variation in ther-mal resistivitywith temperatureanddoseis illustrated in Fig.1. The datawasobtainedfromthe nuclearexperimentswith equiv-alent DIDO nickel dose (EDND). The neutron dose is expressed in terms of equivalentDIDO nickel dose(EDND) which is calcu-lated so that the damage rate experienced by the graphite at a given location is equivalent to that of graphite test samples ir-radiated in the DIDO reactor [17]. 1n/cm−2 _{EDND is equal to}

1_.313 _×10−21 _{displacementperatom(dpa)[12].}

Using thedata ofthermal resistivity fromFig.1, an empirical equationshowninEq(3)forthedependenceoftheresistivityon dose(D)andtemperature(T)isfitted.

K = Ko.

(

a.D+ b

)

×exp

_c

T

+ 1 −

1

(3)

where K is the thermal conductivity of the nuclear graphite, D is dose×1020 _n/cm−2 _{EDND, T is the irradiation temperature in}

Kelvin and Ko is the initial thermal conductivity of unirradiated

graphiteat30°C (about130W/m/K), anda,bandc are constants equalto0.004185,0.9716and209.4,respectively.

Please note thatthe datafor25°C wasexcluded fromthe fit-tingprocess toreduce the errorsumofsquares(SSE)from3.742 to1.858,withcoefficientsof95%confidencebounds.Theresultant

Eq(3) canbe used to calculatetransientchange inthermal con-ductivityforthethermalFEmodelforwhichchangesindoseand temperatureareexpected.Finally,toillustratetheaccuracyofthe calculated thermalresistivity usingEq (5),the valuesare plotted againstexperimentalresultsasshowninFig.2at100°C,300°Cand 600°C.

3. Graphitebrickmodel

3.1. Geometryandmeshoffiniteelementmodelofgraphitebrick

In this paper,failure of the graphite brickof Gilsocarbon un-der irradiation and temperatureis estimated. Inprevious studies

Fig.2. Thermalresistivityvariationwithfastneutrondoseasafunctionofdose andtemperatureasfittedbytheproposedformulacomparedtoexperimental re-sultsfrom[12](for100°C,300°Cand600°C).

within the brick.Theempirical equation (presented inSection 2) oftheevolutionofthethermalconductivityasaresultof irradia-tionandtemperaturechangeisusedforthethermalmodel. Resul-tantnodaltemperaturedistributionisimportedtothemechanical FEmodelforstressandfailureanalyses.

TheFEmodelisdevelopedherebasedonthedimensionsofa particular brick model used by Tsang and Marsden [2]. In order to save computation time, only one-eighth of hypothetical-brick design wasmodelled, asshownin Fig. 3.The fillet radius atthe cornerofthekeywayis6mm.Appropriatesymmetryand circum-ferential periodicboundaryconditionswere appliedalongB1 and B2(see. Fig.3(a))asdescribedby Zouetal.[24].The FEmeshof thebricksectionusedforboththermalandmechanicalanalysesis showninFig.3(b).Themeshisrefined atthestress concentration region (at the keyway fillet), butlarger element sizeswere used forregionsfarfromthatinterest,asshowninFig.3(b).

3.2. Thermalfiniteelementmodel

TheheattransferanalysisfromthethermalFEmodelprovides thetemperaturedistributionnodalfield variablesacrossthe brick which can be implementedinto the mechanicalFE modelasthe

Fig.4. SchematicillustrationofathermalFEmodelofthegraphitebrick.

predefinednodalfields.Fig. 4showsaschematicillustrationof a thermalFEmodelofthegraphitebrick.Asteady-stateheat trans-ferwasappliedwithinner andouter temperatureboundary con-ditionsonthebricksurfaces, T1=500°CandT2=400°C,

respec-tively. Besides, the convection heat transfer from the inner and outer surfaces can be simulated by using the coefficient of heat transfer of the inner (h1) and outer (h2) surfaces assuming that

thecoolantisCO2.Theinitialthermalconductivityofthebrickis

takenas130W/m/K[18]. The plane strain elementsoffour-node lineardiffusiveheattransfer(DC2D4)wereused.

FortheparticularsectionofthebrickshowninFig.4,the gov-erningequationsoftheheattransferare:

Q =

T RTotal

=

(

T2−T1

)

RTotal

(4)

RTotal= RInner+ RBrick+ ROuter (5)

RInner = 1 h1.A

(6)

RBrick= X

KA (7)

ROuter= 1 h2.A

(8)

where Q is the heat flux W/m2_{, T is the temperature, R} Total is

thetotalthermalresistance,ROuter istheconvectionresistance

be-tween the outer surface of the graphite brick and the coolant,

Fig.5. FlowchartrepresentingFEmodelwiththemodelofUMATandfracture.

RInner istheconvectionresistancebetweentheinnersurfaceofthe

graphitebrickandthe coolant,RBrick is theconductionresistance

within the brick,A is the area of conduction or convection (as-sumedidentical andunity) and X is theradial distance between theinnerandoutersurfaces.

R_inner and Router are not considered forthe currentmodel to

avoida complexforcedcooling model offlowingCO2 coolant. In

contrary, nodes at the inner and outer surfaces of the brick are fixedat500°C and400°C,respectively,by assuming that theCO2

coolantwill keepthis temperaturedifference atthe steadystate. Thereforethechangein thetemperaturewithin thebrickwillbe givensolely bythechangein R_Brick duetothechangeinits con-ductivity,K.

The doseis implemented asone of the field variables of the thermalmodel.Itisrampedupwardsfromzeroatthebeginningof thereactorlife.Attheendofoperation(30years)thedosevaries quadraticallyacross the brickfrom 214×1020_n/cm2 _{atthe inner}

radiusto106×1020_n/cm2 _{attheexterior.Withthe known}

infor-mationofdoseandtemperature, updatedthermal conductivityis calculatedusingEq(3)andstoredasthefield variableswhichare implementedinthemechanicalFEmodel.

3.3.Mechanicalfiniteelementmodel

With regard tothe mechanicalmodel,a user-definedmaterial modelforthe constitutive stress/strain behaviour of thegraphite from Tsang and Marsden [2] was used. In the material model of graphite brick, different physical effects such as changes of graphitedimension,porosityandmicrostructurefromtheeffectof irradiation,radiolyticoxidationandfastneutrondamagearetaken intoaccount.Thismodel wasthen adapted toinvestigatethe be-haviourofthe brickfracture usinga traction–separationcohesive modelin[1].TheFEsimulationwassetfor30yearsofplant oper-ation.Inthispaper,temperaturefieldiscalculatedfromthe sepa-ratethermalFEmodelanditseffectsoncrackinitiationand prop-agationwithinthegraphitebrickwasinvestigated.

AnABAQUSsolverwithausermaterialsubroutine(UMAT)was usedto calculate theresultant stresses in themechanical model.

Fig. 5 summarises the mechanical FE model based on four field operating variables. In each analysis, four field-variables profiles

(the operating and irradiation temperatures,neutron fluence and weight loss) are implemented toresemble thecondition of reac-tor operation. The operating temperature is the same as the ir-radiation temperature duringsteady state whereas the operation temperatureisrampeddown totheroom temperatureatcooling down stepafter operating.The nodaltemperaturefromthe ther-mal analysisis implemented into themechanical model.Fluence (radiationdose) variationismentionedinSection 2.Anempirical relationshipfromEasonetal.[18]wasusedfortheweightlossand itcanbeexpressedasafunctionoftheradiationdoseasshownin

Eq.(9).However,theweightlossresultingfromthethermal oxida-tionwasneglectedasthermaloxidationisnotsignificantforlower temperatureCO2cooledUKgraphitemoderators.Itcouldbe

signif-icant for newhigher temperatures moderators. The coolant used inthesimulationwasCO2coolant,asradiolyticoxidationismore

importantthanothertypes,butmodificationstothematerialdata gainedfromTsangandMarsden[20]areneededforfuture gener-ationreactors.

W

(

Dose

)

= 0 .1537 ×Dose (9)

Atthebeginningofeachincrement,ABAQUScalculatesthe to-talstrainwithinthemodel.Thetotalstrainisdecomposedinto es-timatedstrain componentsrelatedtodifferentphysicalprocesses. There are seven strain components decomposed from the total strain(ɛ T_{),asshowninEq(10).}

ε

T₌

ε

e₊

ε

pc₊

ε

sc₊

ε

dc₊

ε

th₊

ε

ith₊

ε

idc ₍₁₀₎

where ɛe_,ɛpc_,ɛsc_,ɛdc_,ɛth_,ɛith_,ɛidc _{are the elastic, primary creep}

(re-coverable), secondary creep (unrecoverable), dimensional change, thermal, interaction thermal and interaction dimensional change strains,respectively.

From Hooke’s law of linear elasticity, the_ɛe _{is calculated. The}

UKAEA model ofKelly andBrocklehurst [19] wasused to model thecreepbehaviour.Theɛpc _{and ɛ}sc _{areirradiationinduced creep}

strains and were assumed to be temperature independent. The ɛ pc _{leads to fast deformation and then gradually decreases with}

increasing irradiation fluence, followed by the ɛsc_{, which is a}

steady state creep strain. The ɛdc_{was assumed to be dependent}

Table1

Graphite tensile-strength valuesatdifferent valuesofdose.

Weightloss(%) Tensilestrength(MPa)

0 22.2

5 42.6

10 36.6

15 31.0

20 26.0

30 17.9

35 14.8

ingraphite. Theseinteractionstrains wereproposed by Kelly and Burchell [20] as a correction factor to accommodate the change made by the total creep strain on CTE of graphite and dimen-sional change strain. These differentstrains create a discrepancy of various stresses that in turn creates a complex stress field. The UMAT then calculates the total stresses, updates them and prepares the Jacobin matrix ((

∂

σ

ij)/(

∂

ɛij)) at the end of the

increment to support convergence of solution. The (

∂

σ

ij) and

(

∂

ɛ ij)are changed in terms of stress and strain, respectively.

ABAQUSFEcodethenusesthecurrentstressesinthelinear dam-age modelto predictinitiation ofcracks andthen their propaga-tion.Based onthestressfield created,crackinitiationand propa-gationcanbeestimatedusingafracturemechanicsbaseddamage model.

3.4. Fracturemechanicsbaseddamagemodel

Graphite is a quasi-brittle material andthe initiation ofcrack is assumedto be driven by thestress states within thematerial. Differentfailurecriteriacanbeusedforcrackinitiation(maximum principalstress,maximumnominalstressetc.).Maximumprincipal stresscriterionisusedinthecurrentworkandforthiscriterion,a crackinitiatesoncethestress withinthesystemexceedsthe ten-silestrengthofthegraphite.Forvirgingraphite,tensilestrengthof 22.2MPa is takenfrom Shiet al.[21]. Due to thelack of tensile strengthevolutiondatawithirradiationdoseandweightloss,the evolutionisassumedtobesimilartothetrendofflexuralstrength givenin[22].Withthisassumption,thetensilestrengthevolution with the weight loss is tabulated in Table 1. Shi et al.[21] rea-soned that the initial increase in strength is the result of hard-ening causedby neutronradiationalthough theoverall reduction in strength is observed as the weight loss becomes higher than 5%.Numerically,extendedfiniteelementmethod(XFEM)isapplied hereforcrackinitiationforthepresentwork.Itallowscrack initia-tionwithinarbitraryelementsoncethefailurecriterionismet.The directionofcrackpathisassumedtobeperpendiculartothe direc-tionofthemaximumprincipalstress andthus thecurrentmodel simulatesModeIcrackgrowth.

Graphite does not fail instantly when the maximum princi-pal stress exceeds tensile strength like pure brittle materials. It hasa strainsoftening region becauseofmicro-crackingandgrain bridgingnearthecracktip.Acohesivetraction-separationdamage model is applied to simulatethis type of behaviour. It is a

phe-nomenological model by combing complex damage mechanisms

aroundthecracktipintoasinglecohesivezone.Theapproachhas beenusedbyother researchers[23,24]forfractureanalysisof nu-cleargraphite.Thetraction stress(T)atthecohesivezone canbe relatedtothecrackseparation(

δ

)asshowninFig.6.Ifgraphiteis a pure brittlematerial, itwill fail whenTexceeds themaximum tractionstress(T),whichisthesameasthetensilestrength. How-evergraphitehasastrainsofteningregiononceTisreachedwhich isillustratedbythelineXZinFig.6.ThegradientofthelineOX(K) representsthe cohesivestiffnessofan undamagedmaterial.Once

Fig.6. Lineartraction-separationcohesivemodel.

the crackinitiates the stiffnessreduces to K(1-D). Adamage pa-rameter(D)is setsothat D=0andD=1representzerodamage atpoint Xandafullycrackedconditionatpoint Zrespectively. It can be proved that Dat pointY (Fig. 6), canbe linked to maxi-mumcrackseparation(

δ

z),separationatmaximumtractionstress

(

δ

), and at the unloading point Y (

δ

Y) as shown in Eq(11). The

criticalcracklength openingisdescribedintermsofthefracture stress(T

)

andthefracturetoughness(Kc)byEq(12).Thearea

un-derneaththetriangleOXZ(Fig.6)representsthecriticalstrain en-ergyreleaserate.Oncethisenergyisachieved,thecohesivezoneis completelydamagedandthecracktippropagatesfurther.Thereis noexperimental dataforthedependenceofcriticalstrain energy on the weight loss due to irradiation dose. Hence it is assumed herethat the criticalstrain energyis dependent only on irradia-tionandthedatafromOuagneetal.[25]areapplied.

D=

_δy

δz

(

δy

−

δ

)

(

δ

z−

δ

)

(11)

δ

z = 2 K2

C

ET (12)

4. Resultsanddiscussion

4.1. Temperaturedistributionthroughthegraphitebrick

ThethermalanalysismodelshowninFig. 4wasimplemented toanalysethe heatflux andtemperaturedistributions within the brick.

Fig.7. Heatfluxcontour(inwatt)through thegraphitebrickwith temperature conditions(500and400°Cattheinnerandoutersurfaces,respectively).

fieldfromthethermalmodelisimplementedintothemechanical model.IncontrarytotheassumptionmadebyTsangandMarsden

[2],anonlinearchangeofthetemperaturealong theradial direc-tionofthebrickisfound.Thisisexpectedsincethesurfacearound thefilletregionisassumedtohavethesametemperature(400°C) astheoutersurfaceofthebrickattheotherregionsforthecurrent model.Hence it has lower temperature than Tsang andMarsden modelforwhichtemperaturevarieslinearlyacrossthebrick. Com-parison of temperature distributions between the current model andthepreviousmodelbyTsangandMarsdenisshowninFig.8. Nodesalong the path B ofthe brick(see Fig.9), have been cho-sentodemonstrate thetemperaturevariationsunder the irradia-tionforthecurrentmodelandTsangandMarsdenmodel.

Although the thermal conductivity could be reduced by as muchas 75%, the updated thermal conductivitydoes not havea noticeableeffectonthe temperaturedistribution.In other words, temperaturedistribution doesnot vary withthe time chosen for thecurrentmodel (30 yearsof irradiation).Thismay be because ofahighinitial thermalconductivityof Gilsocarbon(130W/m/K) andalsoduetotheshortradialdistance(150mm)ofthegraphite brick(shown as X in Eq (7)) through which heat is transferred. Nevertheless, the proposed method for the evolution of thermal conductivitycould be importantinother aspects of heattransfer analyses.Thismayincludeadetailedtransientheattransfer anal-ysisforstart-stoploadcyclesorthecasewhereheatisconducted throughalargerdistanceofthegraphitebrick(axiallengthofthe brick).

4.2.Stressdistributioninthegraphitebrick

Since themaximumprincipal stresscriterion isused forcrack initiation within the graphite brick, it is worthwhile to investi-gatethedistributionofstressesacrossthebrickbefore implement-ing thedamage model. Cracksare expected toinitiate at the re-gionwherepeak maximumprincipal stressesoccurs. The current work simulatedthe stresses through the brickunder two condi-tions;theradialandnon-radialtemperaturedistributionsasshown inFig. 8. From both temperaturedistributions, maximum princi-pal stress concentration can be found at the fillet corner of the graphite brickat the end of 30 year of service life as shown in

Fig.10.Hence,thecrackinitiationisexpectedatthefillet corner. Forbothtemperaturedistributioncasesthereisnosignificant dif-ferenceincontoursforthemaximumprincipalstress(seeFig.10).

The evolutions ofprincipal stresses for both temperature dis-tributioncasesatthefillet cornerduringoperationareplottedin

Fig.11.There isan increase intheprincipal stressatthe startof operation forboth casesuntil thestress stabilises after 5yearof operation. Sudden increase in stresses after the steady state can be found at around 20 year life. The sudden increase in maxi-mum principal stress can be relatedto the increase in the hoop strainrelatedtodimensionalchangeofgraphiteathigherdoseas mentionedbyWadsworthetal.[1].Finalincreaseintheprincipal stressduringthecoolingstage afteroperatingfor30yearsisdue tothemismatchincoefficientofthermalexpansion(CTE) through-out the brick.Ingeneral, higher maximumprincipal stress is ex-pected for a linear temperature distribution cases indicating the crackinitiationtime forthecasecouldbe shorterthanthemodel withnonlineartemperaturedistribution.

4.3. Damageanalysisofthegraphitebrickundertemperatureand

radiationloads

According to the stress analysis from the previous section, cracksareexpectedtoformatthefilletcornerduetothe concen-tration ofmaximumprincipalstress. Thecrack initiationtimefor lineartemperaturedistribution(22yearsand7months)isshorter than thecases withnonlinear temperaturedistribution (23 years and2 months)because of highermaximum principal stress. The finding showstheimportance ofthe temperaturedistributionfor thelifetimeestimationofthe graphitebricks.The typical geome-tryofthecrackinthebrickafter30yearoperation forlinearand nonlineartemperatureprofilesareshowninFig.12(a)and(b), re-spectively. Firstly, thecrack propagatestowards theinner surface andthenchangesitsdirectionwhenreachesadistanceof6.5mm. Thedirectionoffurthercrackgrowthisperpendiculartothehoop direction indicating that the maximum principal stress is in the directionofhoop whenthecrackreaches6.5mm.Thetotalcrack length forbothcases attheendofoperation isaround 65.4mm. When crackgrowthrateis observed,itisshownthat thegrowth ratesforbothtemperaturedistributioncasesaresimilarasshown inFig.13.Rapidcrackgrowthoccurswithin1yearfromcrack initi-ationtime.Becauseofthiscrack,stressesarerelaxedandthecrack lengthreachestheplateaustageafteroneyearfromtheinitiation time.Furtherincreaseincracklengthisfoundaftercoolingdown stage.Duringtheoperation,thebrickisexposedtodifferent tem-perature and radiation dose. Hence, difference in CTE across the brickisexpectedandthisCTEmismatchcausesstresseswhenthe coolingisapplied.Thismismatchstressfurtherincreasesthecrack length by 1.5mm. The maximum principal stress distribution at the endof cooling down for both temperaturedistribution cases areshowninFig.14.

5. Conclusionsandfuturework

Theempiricalformulaofvaryingthermalconductivityof com-mon irradiated graphite form, Gilsocarbon, due to the change in temperatureanddosehasbeendeveloped.This formulais devel-opedbyusingtheexperimentalthermalconductivityofirradiated graphite fromtheliterature. The resultant formulacan be imple-mentedintoeitherthermalFEmodelorcorrespondingmechanical FEmodelandlifeestimationmodels.

(a) (b)

Fig.8. Thecontouroftemperaturedistributionindegreefor(a)TsangandMarsden[2]model,and(b)thethermalanalysismodelofthegraphitebrickofthecurrentwork.

400 410 420 430 440 450 460 470 480 490 500

0 0.2 0.4 0.6 0.8 1

T

emperatur

e(˚C)

Normalised distance (R/Ro)

Tsang and Marsden

Current model

Path B Path B

[2]

Fig. 9. Distribution of the temperature of the nodes along the path B of the graphitebrick(Radialdistancefromthecentreofthebrick(R)isnormalised us-ingtheouterradiusofthebrick(Ro )).

throughthe thicknessoflarger graphitebricks.Thethermal anal-ysisalso showedthat the temperaturedistributiondoesnot vary radially withinthegraphitebrickduetotheeffectofthethermal concentrationinthefilletcorner.

Fig.11. Maximumprincipalstresses(MPa)attheexpectedpointofcrackinitiation (keywayfillet)after30yearofoperationfortwotypesoftemperaturedistribution.

Basedonthetemperaturedistributionobtainedfromthe ther-malanalysis, stressanddamageanalysisofthegraphitebrick un-derneutrondoseandtemperaturewere carriedout.Thedose in-creasesthe weightloss resultingina decrease inthestrength of thebrickaheadthemaximumprincipalstressandthusleadingto

(a) (b)

Fig.12. Typicalcrackgeometryandmaximumprincipalstressdistributionofthebrickafter30yearsofoperationfor(a)linearand(b)nonlineartemperaturedistributions.

Fig.13. Cracklengthduringoperationforthecaseswithalinearandanonlinear temperatureprofiles.

crackinitiationwithinthe brick.Theinitiationofcracksoccursat thefilletcornerofthegraphitebrickanditgrowstowardsthe in-ner bore ofthe brick. The maximum principal stress islower at theexpected crackinitiation region forthe casewhere the

tem-perature distribution is nonlinear compared to the casewith as-sumed lineartemperaturedistribution. Hence the crackinitiation forthecaseisdelayedbyabout7months.Regardlessofcrack ini-tiation time, both cases have similar crack growth rate and the crack growth stops after one year from initiation time. Further crackgrowthisonlydrivenfromCTEmismatchstressesduringthe coolingdownstage.

Thestudyshowstheimportanceoftemperaturedistributionon thelifetimeestimationofgraphitebricks.Subtledifferencein tem-peraturedistribution(asshownbytwodifferentcasesinthe cur-rentwork)couldcausethedifferenceinthecrackinitiationtime. However, there are some areas of the current work that needs tobe improved.A transientheatanalysisconsidering the flowof thecoolantmaybeimplementedinfutureinsteadofusinga sim-pletemperatureboundaryconditionsforinnerandouter surfaces of the graphite brick. In addition, the usage of a more realistic 3D geometry of the brick with cut-out holes for the circulation ofthe coolantmaybe considered inorder topredict more accu-rate transient heat flow and the temperature profiles. Moreover, changes inporosity andphasesin graphiteasdescribedby Kyaw et al. [26] could affectits mechanical properties andthis model could be coupledto the current mechanical modelin the future work.

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