J+Nuclear+Materials+2006+Tsang+208

Journal of Nuclear Materials 350 (2006) 208–220

www.elsevier.com/locate/jnucmat

The development of a stress analysis code for nucleargraphite components in gas-cooled reactors

D.K.L. Tsang *, B.J. Marsden

Nuclear Graphite Research Group, School of Mechanical, Aerospace and Civil Engineering, The University of Manchester,

P.O. Box 88, Manchester M60 1DQ, UK

Received 5 October 2005; accepted 30 January 2006

Abstract

Most of the UK nuclear power reactors are gas-cooled and graphite moderated. As well as acting as a moderator thegraphite also acts as a structural component providing channels for the coolant gas and control rods. For this reason thestructural integrity assessments of nuclear graphite components is an essential element of reactor design. In order to per-form graphite component stress analysis, the definition of the constitutive equation relating stress and strain for irradiatedgraphite is required. Apart from the usual elastic and thermal strains, irradiated graphite components are subject to addi-tional strains due to fast neutron irradiation and radiolytic oxidation. In this paper a material model for nuclear graphite ispresented along with an example of a stress analysis of a nuclear graphite moderator brick subject to both fast neutronirradiation and radiolytic oxidation.� 2006 Elsevier B.V. All rights reserved.

1. Introduction

Most of the UK operating nuclear reactors aregraphite moderated and carbon dioxide cooled.For the reactor designer there are two main advan-tages for the use of gas as a coolant. Firstly thedensity of gas can be varied, allowing the designoperating temperature to be chosen independentlyof the operating pressure. Thus a high gas tempera-ture can be defined and an optimum pressure can

0022-3115/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.jnucmat.2006.01.015

* Corresponding author. Tel.: +44 1612754391.E-mail address: [email protected] (D.K.L.

Tsang).

then be selected separately, accounting for reactorsafety and the economics of pumping power as wellas pressure circuit costs. Secondly reactor operatingconditions can be chosen such that the gas canundergo no phase change as a result of risingtemperature or falling pressure, so gas cooling canbe maintained even under fault conditions. Alsogas flow and temperature can be simply and confi-dently predicted [1].

The early reactor designs [2] in the UK, e.g.Gleep, Bepo and Windscale piles, were graphite-moderated air-cooled reactors. Later, eight dual-purpose reactors (electric generation and plutoniumproduction) were built at Calder Hall and Chapel-cross which used carbon dioxide as a coolant. Thesimple well thought out design of these gas-cooled

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D.K.L. Tsang, B.J. Marsden / Journal of Nuclear Materials 350 (2006) 208–220 209

graphite moderated reactors gave nearly 50 years ofreliable operation. A rapid increase in electric powerdemand in the 1960s led to an increased programfor power generation reactors. Nine Magnox sta-tions were eventually built in the UK with a totalnet output of nearly 5000 MW(e).

The use of magnesium alloy fuel cladding andnatural uranium metal fuel in Magnox reactors ledto relatively low fuel ratings and outlet temperatures�360 �C, which meant that the plant tended to belarge, and costly with relatively low efficiency.Therefore, a better and more advanced design wasdeveloped, the advanced gas-cooled reactor(AGR). The AGR design was based on a slightlyenriched uranium dioxide fuel, encased in stainlesssteel cladding and arranged in clusters of smalldiameter pins. This fuel has the advantage that itis chemically inert in carbon dioxide up to very hightemperatures. In addition the AGR operates athigher coolant pressure than the Magnox reactorsgiving a higher outlet temperature of �650 �Cmaking the steam generation side more efficient.

The high temperature reactor (HTR) was thenext step in the evolution of graphite moderatedgas-cooled reactors. The change in coolant tohelium opens up new possibilities in fuel designand higher operating temperatures. The HTRpromises lower costs through higher core power,smaller plant size and replaceable moderator. Theall ceramic fuel and helium coolant allow for a highoutlet temperature of �1000 �C and lead to thepossibility of using the heat directly for industrialprocesses. Interest was aroused around the worldand led to the construction of experimental and pro-totypic reactors, e.g., DRAGON in the UK, PeachBottom and Fort St. Vrain in the USA, and AVRand THTR in Germany. Interest in the HTRsystems has continued to the present day in manycountries for example the pebble bed modular reac-tor (PBMR) in South Africa, the high temperatureengineering test reactor (HTTR) in Japan and the10 MWt high temperature gas-cooled demonstra-tion reactor (HTR-10) in China. The very hightemperature reactor (VHTR) is now one of the mostpromising designs being considered by the interna-tional Generation IV initiative. One of the mostexciting applications of the VHTR is the efficientproduction of hydrogen by the sulphur–iodinecycle.

In a gas-cooled reactor the properties of thegraphite components are changed by fast neutronirradiation and, in the case of air or carbon diox-

ide-cooled reactors, by radiolytic oxidation. Theseirradiation induced changes can lead to the buildup of significant stresses and deformation in thegraphite components. Throughout reactor life it isessential that the graphite core structure remainssufficiently strong and undistorted in order to main-tain fuel cooling, permit loading and unloadingof the fuel and allow the necessary movements ofcontrol rods, in both normal and fault conditions.

In order to perform structural integrity assess-ments, the definition of the constitutive equationrelating stress and strain in the irradiated graphitematerial is required. Apart from the elastic andthermal strain, graphite in nuclear reactors alsoexperiences additional strains due to fast neutronirradiation. Irradiation creep and irradiationinduced dimensional change are two most impor-tant of these irradiation strains. In some cases, irra-diation creep strain is beneficial because of stressrelief, especially in regions of high stress concentra-tions. Although the irradiation creep strain relaxesstresses generated by temperature gradients andthe irradiation induced dimensional changes tosome extent, significant component stresses can stillbe generated. Furthermore, the dimensional changestrain in graphite components can lead to similarinternal stress distributions to those generated bythermal loading and these additive stress distribu-tions may be critical in design. Therefore, it isimportant to consider the effect of irradiation creepstrain, thermal strains and dimensional changes ingraphite component stress analysis.

Nuclear graphite component structural integrityassessments are usually carried out using the finiteelement method. In order to model the dimensionalchange, irradiation creep and material properties acomplex material model is required. The modelsused are largely empirical and are based on materialtest reactor (MTR) irradiated graphite propertydata. The application of nuclear graphite constitu-tive equations using standard finite element tech-niques are further complicated as irradiation creepin graphite is considered not to conserve volume[3], i.e. with irradiation, the volume of graphiteincreases under tension loading and decreases undercompression loading. Consequently, specially writ-ten coding is required to model the behaviour ofirradiated graphite components.

A viscoelastic graphite model was first producedin the USA by Chang and Rashid [4]. Later, amodel called VIENUS was created by JAERI inJapan for a HTTR graphite block by Iyoku et al.

210 D.K.L. Tsang, B.J. Marsden / Journal of Nuclear Materials 350 (2006) 208–220

[5]. In the UK, a model for two dimensions and axi-symmetric geometries was created using BERSAFEfinite element code by Harper [6]. This was laterdeveloped using the ABAQUS [7] finite elementprogramme by Whittaker [8] using a User MATerial(UMAT) subroutine to model Gilsocarbon graphitecomponents in two dimensions.

A material model subroutine called MANUMAT has been recently developed for nucleargraphite under fast neutron irradiation and radio-lytic oxidation conditions. The subroutine can beused together with the ABAQUS finite elementprogram to perform three-dimensional, time-inte-grated, non-linear irradiated graphite stress analy-ses. The subroutine has been used to study thegraphite brick bore profile [9]. In this paper, thedevelopment of the MAN UMAT is presented.

2. Unirradiated nuclear graphite material

properties

The mechanical properties of isotropic graphiteare given in Table 1. The density of pure graphitecrystals is 2.26 g/cm3, however the density of the

Fig. 1. Stress–strain curves for unirradiated

Table 1Typical virgin material property for isotropic graphite

Material property ofisotropic graphite

Values

Density 1.81 g/cm3

Mean coefficient ofthermal expansion (CTE)

4.35 · 10�6 (�C)�1

Poisson’s ratio 0.2Dynamic Young’s modulus 10 GPaRatio of SYM to DYM 0.84

manufactured graphite has a much lower densityas can be seen from Table 1. The lower density isdue to gas evolution porosity and inter-crystallinecracks formed during the baking and graphitisationstages of graphite manufacture. The inter-crystallinecracking which gives graphite its unique unirradi-ated and irradiated properties is often referred toas Mrozowski cracks [10]. Mrozowski first proposedthese cracks and suggested models for the edge link-ing of mutually oriented crystallites. He attributedthe internal voidage to the basal plane micro-cracksproduced during the anisotropic shrinkage of thecrystallites on cooling from graphitising tempera-tures.

In the UK nuclear industry, the coefficient ofthermal expansion (CTE) is usually quoted in therange 20–120 �C although measurement over a lar-ger range, say 20–400 �C yields more accurate value.The CTE is temperature dependent with an increasein temperature producing an increase in CTE. Pois-son’s ratio for elastic strain is assumed to be 0.2 andto be independent of irradiation.

The stress–strain curves for unirradiated isotro-pic graphite subjected to cyclic loading are shownin Fig. 1. When an external load is applied, the ini-tial macro-deformation is governed by elastic shearand plastic deformation of individual crystallites,the latter causing a non-linear macro stress/strainresponse. A large permanent set occurs when theload is removed.

Most of the Young’s modulus data are obtainedby an ultrasound method. The method representsthe slope of the stress–strain curve at the origin torepresent the modulus and the modulus is referredto as the dynamic Young’s modulus (DYM) Ed.

graphite subjected to cyclic loading.


The Young’s modulus can also be obtained in ‘sta-tic’ tests using strain gauges. This value is obtainedfrom the average slope of an unload–load cycle ofthe stress–strain curve to define the recoverable elas-tic strain and is referred to as the static Young’smodulus (SYM) Es. In the UK nuclear industrythe stress load level has been traditionally set tobe less than or equal to half the tensile strengthand hence does not include the permanent set strainobserved in general static tests. The mean ratio ofSYM to DYM is about Es/Ed = 0.84 for unirradi-ated graphite depending on the load range overwhich it is measured. More information about rela-tion between SYM and DYM of nuclear graphitehas been published by Oku and Eto [11].

3. UK methodology for isotropic graphite

irradiated in CO2

There are two processes in nuclear reactors whichmodify the graphite properties significantly. Thefirst process is the creation of lattice defects bythe energetic radiation, principally the neutrons.The neutrons emitted in the fission process in thefuel of a thermal reactor lose their excess energy incollisions with the nuclei of the moderator atoms.A primary atomic displacement produced by a neu-tron–nuclei collision with a carbon atom produces acascade of further displacements. Detailed calcula-tions have been performed by a number of authors[12,13]. The calculations indicate that 200–1000atoms are displaced by the most energetic collisions,spread out over a large volume, such that it is agood approximation to consider that the displace-ments occur at random. During irradiation, crystalinterstitials and vacancies are formed when fast neu-trons displace carbon atoms from their lattice posi-tions. These point defects can recombine to form avacancy or interstitial loop. The rate and extent towhich the recombination takes place is dependenton the irradiation temperature. Measurements ofthe crystal lattice parameters using X-rays showedthat the interlayer spacing increases while theatomic spacing in the layer decreases under irradia-tion. Direct transmission electron microscopy byReynolds and Thrower [14] showed that the crystalgrowth is due to the formation and growth of inter-stitial dislocation loops. The large basal plane con-tractions can be explained by the collapse of linearvacancy groups, the two-dimensional analogue ofthe vacancy loop. In short, neutrons bombardmentof graphite initially creates point defects, which,

depending on the irradiation temperature lead tomore complex defects with increasing irradiationfluence. The lattice defects lead to the changes inthe properties of the graphite.

The second process is radiolytic oxidation. Underfault conditions graphite oxidation by thermalreaction can predominate in the high temperaturegas-cooled reactor. However, thermal oxidation isinsignificant in the lower temperature CO2-cooledgraphite moderators. Radiolytic oxidation occurswhen CO2 is decomposed by ionising radiation togive reactive oxidising species. The rate of graphiteoxidation is proportional to the rate of productionof the oxidising species, which is itself a functionof radiation intensity and gas pressure (but is sub-stantially independent of temperature). Reviewswhich summarise the relevant information on oxi-dation of graphite due to radiolytic activation ofcarbon dioxide and the combination of the effectsof oxidation with those of irradiation damage topredict the operational life of the graphite modera-tor can be found in Best et al. [15] and Kelly [16].

In this section changes in the structural, mechan-ical and thermal properties of isotropic graphite dueto fast neutron irradiation and radiolytic oxidationare given. Property variations are given as a func-tion of fast neutron fluence, temperature, fractionalweight loss and graphite oxidation rate. These fieldvariables are calculated using other analysisprograms. The units for irradiation fluence andirradiation temperature are expressed in terms ofequivalent DIDO nickel dose (EDND) and �C.

3.1. Young’s modulus

Following irradiation the stress–strain relation-ship become more linear and the permanent set isdecreased. Experiments to determine the elasticresponse of irradiated graphite have tended toconcentrate on the dynamic Young’s modulus asthe basic of measurement. At low irradiation fluencelevels there is a rapid increase in modulus followedby a constant state. At higher fluence levels theYoung’s modulus shows a gradual increase to apeak followed by a fall (see Fig. 2).

There are two distinct mechanisms involved inthese changes, firstly due to changes in the elasticconstants of the graphite crystals and secondly dueto changes in the bulk structure of the graphite.The first of these which is responsible for the initialrise in Young’s modulus, is attributed to the pin-ning of the crystal basal planes and this effect soon

Fig. 2. Young’s modulus of an irradiated isotropic graphite attemperature 550 �C.


saturates. The second, is attributed to differentialstraining of the polycrystalline micro-structure dueto crystal growth and porosity closure leading toan increase in modulus. However, at even large crys-tal strains micro-cracks are generated which eventu-ally leads to the reduction of the modulus and lossof strength.

The irradiated DYM Ed can be described by thefollowing equation:

Ed

E0

¼ P ðT Þ � Sðc; T Þ � W ðxÞ; ð1Þ

where E0 is the unirradiated DYM, P is the satu-rated pinning term to account for the pinning ofmobile dislocations and is assumed to be a functionof temperature only, S is the structure term to allowfor structural changes due to irradiation damageand is a function of fluence c and irradiation tem-perature T, and W is the weight loss term. Theweight loss term is given as a function of fractionalweight loss by [16]:

W ðxÞ ¼ expð�k1xÞ; ð2Þwhere x is the fractional weight loss and k1 = 3.4 isan empirical constant. The irradiated SYM is givenby

Es

E0

¼ fi � P ðT Þ � Sðc; T Þ � W ðxÞ; ð3Þ

where fi is a mean ratio of the irradiated SYM toDYM.

The pinning term is assumed to saturate instanta-neously at low fluence. It is also assumed that thepinning term and the structure term can be calcu-lated separately and combined using the multiplica-tion law given in Eq. (1) above. These assumptionsare required as it is considered that the pinning termdoes not affect irradiation creep but the structural

changes do. It is assumed that Poisson’s ratio isnot a function of temperature, fluence or weightloss, and stays constant at a value of 0.2.

3.2. Irradiation creep

The fact that graphite undergoes irradiation-induced creep is important because, as graphitemoderator blocks in a nuclear reactor are subjectedto significant temperature and irradiation fluencegradients, and irradiation induced dimensionalchanges in graphite are both temperature and flu-ence dependent, these lead to self-induced stressesto be imposed on the blocks. These stresses wouldbuild up to levels exceeding the fracture strengthof graphite [17] in many cases if these stresses werenot relieved by irradiation creep [18].

The creep coefficient is considered to be equal intension and compression for a given graphite. Irra-diation creep strain is defined as ‘the difference indimensions between a stressed sample and a samplewith the same properties as the stressed sample irra-diated unstressed’ [19]. It has been postulated thatthe irradiation creep in graphite is due to slip inthe basal plane activated by the neutron irradiationand not by internal stress. The creep rate is consid-ered to be proportional to stress and practicallyindependent of temperature in the range 300–650 �C.

Irradiation creep can be characterised by atransient stage followed by a linear stage. The mag-nitude of both primary transient creep and second-ary creep was found to be proportional to stress andinversely proportional to the unirradiated modulusof elasticity [20].

3.3. Coefficient of thermal expansion

Fast neutron irradiation fluence modifies thecoefficient of linear thermal expansion (CTE). Thischange is a function of the irradiation fluence, irra-diation temperature and actual temperature. How-ever, graphite CTE does not appear to be affectedby radiolytic oxidation [16]. The CTE is used toevaluate thermal strain and consequently the calcu-lation of its value involves a change in temperatureor in CTE. The instantaneous coefficient of linearthermal expansion ai at temperature T can bewritten as

aiðT Þ ¼deth

dT; ð4Þ


where the subscript i represents instantaneous value.Therefore the thermal strain can be written as

ethðT Þ ¼Z T

T ref

aiðhÞdh; ð5Þ

where h is a dummy variable of integration and Tref

(�C) is a reference temperature. As the CTE ofgraphite is temperature dependent, most practicalapplications require the mean CTE over a tempera-ture range. For convenience it is useful to introducethe reference temperature Tref = 20 �C and refer tothe mean CTE over a temperature range:

�aðT ref –T Þ ¼1

ðT ref � T Þ

Z T

T ref

aiðhÞdh; ð6Þ

where �aðT ref –T Þ is the mean CTE over the temperaturerange Tref–T �C for either unirradiated or irradiatedgraphite. The mean CTE over any desired tempera-ture range T1–T2 �C can then be expressed as

�aðT 1–T 2Þ ¼1

ðT 2 � T 1Þ½ðT 2 � T refÞ�aðT ref –T 2Þ

� ðT 1 � T refÞ�aðT ref –T 1Þ�. ð7Þ

Using Eq. (6), the thermal strain caused by the tem-perature difference from Tref to T can be rewritten as

eth ¼ �aðT ref –T ÞðT � T refÞ. ð8Þ

As is the case for unirradiated CTE, irradiatedCTE is usually quoted over the temperature range20–120 �C in the UK. However, the irradiated meanCTE over some other temperature range 20–T �Ccan be written as [21]:

�að20–T Þ ¼ �að20–120Þ þ B; ð9Þ

where B is a function of the actual temperature Tand the irradiated mean CTE �að20–120Þ is a functionof irradiation fluence and temperature. The experi-mentally determined irradiated mean CTE can beobtained from measurement made on graphite sam-ples irradiated in a MTR.

Early irradiation creep experiments [22] haveshown that changes to the thermal expansion coeffi-cient in irradiated specimens appear to be depen-dent on the magnitude of creep strain. Thosespecimens with a significant amount of compressivecreep strain have a higher CTE, measured in thedirection of stress, compared with the unstressedspecimens, whereas specimens with significanttensile creep strain have a lower thermal expansioncoefficient compared to the unstressed specimens.

3.4. Dimensional change

Radiation damage by fast neutron fluence causescrystal growth in the c-axis direction and shrinkagein the a-axis direction [23]. The effect of these irradi-ation induced crystal changes on the dimensionalchanges in isotropic moderator graphites has beenreviewed by Neighbour [24]. Neutron irradiation ofpolygranular reactor graphites results in initial bulkshrinkage at low neutron fluence, leading eventuallyto a net expansion of the graphite with increasing flu-ence. Above irradiation temperatures of 300 �C, theinitial effect of neutron irradiation on the microstruc-ture of the graphite is closure of small pores andcracks as a result of c-axis expansion and contractionof the crystallites in the a-axis causing the initial bulkshrinkage of the graphite. The reversal of shrinkage,called ‘turnaround’, is believed to begin when theshrinkage cracks are unable to accommodate newirradiation-induced crystallite growth. At higherirradiation temperatures, turnaround occurs atlower neutron fluence. This is attributed to a reduc-tion in the extent of accommodation available forirradiation-induced c-axis expansion as a result ofclosure of Mrozowski cracks by thermal expansion.With further crystallite growth after turnaround,internal stresses develop to the point where newmicro-cracks and pores begin to appear.

The amount of dimensional change is assumed tobe a function of irradiation fluence and irradiationtemperature. However, there are little experimentaldata [25] for the combined effect of the irradiationand radiolytic oxidation in a CO2 atmosphere. Toenable structural integrity calculations to be carriedout it has been assumed that for a given temperature,the dimensional change against fluence can be repre-sented by the linear combination of two curves [16]:

• Curve A: maximum effect of the weight loss. Thecurve continues to shrink over the full fluencerange. Curve A is obtained from theoretical con-sideration of the relationship between crystal andpolycrystalline dimensional change rates andCTE [13], and data obtained on graphite crystalstructures [23].

• Curve B: no weight loss. Curve B is an empiricalcurve obtained from graphite irradiation experi-ments. At low fluence Curve B follows CurveA, but gradually the rate of shrinkage decreases,eventually passing through a maximum shrink-age value before reversing, often referred to as‘turnaround’.

Fig. 3. Schematic of a dimensional change with radiolyticoxidation.


For a given fluence and temperature, the dimen-sional change strain can be determined by linearinterpolation between the positions on Curve Aand Curve B (see Fig. 3):

edc ¼ /� edcA þ ð1� /Þ � edc

B ; ð10Þwhere edc is the dimensional change strain, edc

A is thedimensional change strain for Curve A as a functionof fluence and temperature, edc

B is the dimensionalchange strain for Curve B as a function of fluenceand temperature, / is the fractional distance be-tween Curve B and Curve A, which is a functionof the graphite oxidation rate.

It should be noted that this is an approximationfor the effect of radiolytic oxidation on the dimen-sional changes of graphite due to irradiation, per-mitting interpolation between two curves, Curve Bcorresponding to zero oxidation rate and Curve Ato oxidation rate sufficiently fast that micro-crackclosure due to neutron irradiation is prevented.However, the actual situation is probably morecomplex and requires further study.

It has been postulated that similar to CTE, theirradiation creep perturbs the graphite microstruc-ture enough to produce significant modificationsto the dimensional change rate (see [19,26]). Acorrection term in the dimensional change has beenproposed by Kelly and Burchell [19].

4. Constitutive equations

The complex material behaviour of irradiatedgraphite has been implemented into the ABAQUS[5] finite element code using a User MATerial sub-routine (UMAT). At the beginning of each incre-ment, ABAQUS estimates the total strain withinthe model. There are two functions performed by

the UMAT subroutine. Firstly, it updates the stres-ses to their values at the end of the increment fromthe estimated total strain. Secondly, it provides thegraphite material Jacobian matrix oDr/oDee forthe mechanical constitutive model, where Dr andDee are the changes in stress and elastic strain atthe end of current time increment, respectively.Since the UMAT requires the increment in stressand elastic strain, the constitutive equations arepresented here in incremental form.

The total strain within the model is the sum ofseven different strain components:

Detotal ¼ Dee þ Depc þ Desc þ Dedc þ Deth

þ Deith þ Deidc; ð11Þ

where Dee is the elastic strain, Depc is the primarycreep strain, Desc is the secondary creep strain, Dedc

is the dimensional change strain induced by irradia-tion, Deth is the thermal strain, Deith and Deidc arethe interaction thermal and dimensional changestrains due to creep, respectively. All of these strainsare described in more detail below.

4.1. Elastic strain

The elastic strain is related to the stress by meansof Hooke’s law of linear elasticity, i.e.

r ¼ Dee; ð12Þ

where the material matrix D is a function of irradi-ation temperature and irradiation fluence. Theincremental equation for Hooke’s law can be writ-ten as

Dr ¼ eD � Dee þ DD � ~ee; ð13Þ

where tilde (�) represents the mean value in currentincrement. It is assumed that there are no bodyforces and the stress satisfies the equation of equilib-rium, i.e.

rij;j ¼ 0 ð14Þ

and the strain–displacement relationships is

eeij ¼

1

2ðui;j þ uj;iÞ; ð15Þ

where the tensor notation has been used and thecomma represents partial differentiation.


4.2. Creep strain

In the UK, total irradiation creep strain has tra-ditionally been separated into two creep strain com-ponents [3]:

ecreep ¼ epc þ esc; ð16Þ

where epc and esc are the primary and the secondarycreep strains, respectively. The primary creep strain,which is also called transient creep, is a recoverablestrain and is defined as

epc ¼ 4ue�4c

Z c

0

Dcre4c0 dc0; ð17Þ

where u(T) is a parameter dependent on tempera-ture T, c is the irradiation fluence and Dc is the usualcreep material matrix in which the creep modulus Ec

is defined as a function of unirradiated modulus,structure term and weight loss as

Ec ¼ E0 � fi � Sðc; T Þ � W ðxÞ

and Poisson’s ratio for creep is taken as being thesame as the elastic Poisson’s ratio and to be invari-ant to fast neutron irradiation and radiolyticoxidation.

The primary creep is characterised by a rapiddeformation which soon decreases in rate withincreasing irradiation fluence. To put Eq. (17) inincremental form, first Eq. (17) is rewritten as adifferential equation

depc

dc¼ 4uðDcr� epcÞ ð18Þ

and hence the incremental equation for primarycreep can be written as

Depc ¼ 4uðeDc~r� ~epcÞDc. ð19ÞThe secondary creep strain, which also called

steady state creep, is an irreversible strain and isdefined as

esc ¼ nðT ÞZ c

0

Dcrdc0; ð20Þ

where n(T) is a temperature dependent parameter.The secondary creep is characterised by a lineardependence of the creep strain. The incrementalequation for secondary creep can be written as

Desc ¼ neDc~rDc. ð21ÞIt is assumed that both parameters u and n are inde-pendent of temperature in this paper.

4.3. Thermal strain

The thermal strain is defined as

eth ¼ �aðT ref –T ÞðT � T refÞ; ð22Þ

where �aðT ref –T Þ is the mean coefficient of thermalexpansion between temperature Tref–T �C, and Tref

is a reference temperature. Differentiating the aboveequation, we have

deth ¼ d�aðT ref –T ÞðT � T refÞ þ �aðT ref –T Þ dT . ð23Þ

Using Eq. (9), Eq. (23) can be simplified to the fol-lowing difference equation:

deth ¼ d�að20–120ÞðT � T refÞ þ ai dT ð24Þ

and hence the incremental equation for thermalstrain can be written as

Deth � D�að20–120ÞðT � T refÞ þ ai DT . ð25Þ

The small orthotropic CTE in semi-isotropicgraphite is also accounted for in the code, i.e.

�aðT 1–T 2Þ ¼�a1ðT 1–T 2Þ

�a2ðT 1–T 2Þ

�a3ðT 1–T 2Þ

0BB@

1CCA; ai ¼

a1i

a2i

a3i

0B@

1CA; ð26Þ

where �a1ðT 1–T 2Þ, �a2

ðT 1–T 2Þ and �a3ðT 1–T 2Þ are mean CTE

along orthogonal axis 1, 2 and 3, respectively andsimilar for instantaneous CTE ai. The mean CTEis a function of fluence, irradiation temperatureand actual temperature. The mean CTE over anydesired temperature range can be found from Eq.(9). Once the values of CTE in different directionsare known, the incremental thermal strain can becalculated from Eq. (25).

4.4. Dimensional change strain

The amount of dimensional change is assumed tobe a function of irradiation temperature fluence,temperature and weight loss, and is assumed to bethe same in all directions. It is assumed that for agiven temperature, the dimensional change againstfluence can be represented by

dedc

dc¼ /

dedcA

dcþ ð1� /Þ dedc

B

dc; ð27Þ

where edcA is the dimensional change strain with

maximum effect of weight loss, edcB is the dimensional

change strain without weight loss effect and / is thefractional parameter between the maximum weight


loss, Curve A, and no weight loss, Curve B. Bothterms edc

A and edcB have been determined, therefore

the termsdedc

A

dc anddedc

B

dc can be numerically calculated.The incremental form of Eq. (27) can be written as

Dedc � /dedc

A

dcþ ð1� /Þ dedc

B

dc

� �Dc. ð28Þ

Hence the change in dimensional change strain canbe calculated.

4.5. Interaction strains

The two interaction strains in Eq. (11) areassumed to be a function of irradiation creep ingraphite. These are the interaction thermal strainand the interaction dimensional strain.

For small creep strain, the correction in irradi-ated mean CTE að20–120Þ can be calculated using fol-lowing linear relationship:

að20–120Þ ¼ jeec; ð29Þ

where

eec ¼

1 �l �l 0 0 0

�l 1 �l 0 0 0

�l �l 1 0 0 0

0 0 0 1þ l 0 0

0 0 0 0 1þ l 0

0 0 0 0 0 1þ l

26666666666664

37777777777775

ecreep1

ecreep2

ecreep3

ecreep12

ecreep13

ecreep23

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

ð30Þis called the effective creep strain, j is the slope atthe origin of the CTE/creep curve and l = 0.5 isthe lateral coefficient. Once the correction in CTEað20–120Þ is found, the incremental interaction ther-mal strain is defined as

Deith � Dað20–120ÞðT � T refÞ þ að20–120ÞDT . ð31Þ

Since the interaction thermal strain is a correction inCTE due to the total creep strain. The expression inEq. (31) is same as the definition of the thermalstrain given in Eq. (25).

We have seen how the total creep strain modifiesthe CTE. However, the dimensional change strainappears to be a function of CTE, therefore creepstrain may be expected to modify the dimensionalchange. Kelly and Burchell [19] proposed an inter-action dimensional change strain is defined as

eidc ¼Z c

0

að20–120Þ

ðac � aaÞ

� �dX T

dcdc0; ð32Þ

where ac and aa are the crystallite CTE in ‘a’ and ‘c’directions, respectively. The function XT is theso-called shape factor for the graphite crystallitewhich is given by the difference in the crystallitedimensional change in the ‘a’ and ‘c’ directionsdefined as

X T ¼DX c

X c

� DX a

X a; ð33Þ

where XT is a function of irradiation temperatureand fast neutron fluence and has been obtainedfrom irradiation experiments on highly orientatedpyrolytic graphite (HPOG). Hence the crystal shapeterm dXT/dc can been found and the incrementalequation for Eq. (32) can be calculated using

Deidc �_að20–120Þ

ac � aa

�dX T

dcDc;

�ð34Þ

where _að20–120Þ is the average value of corrected CTEat current increment.

The CTE correction term að20–120Þ is a function ofthe total creep strain. Since both the interactionstrains are functions of the CTE correction term,both interaction strains are implicit functions ofthe total creep strain.

5. Numerical technique

In this section, the implementation of the pro-gram is described. The material model has beenimplemented into ABAQUS [7] via user materialsubroutine (UMAT). If interaction strains are notincluded in an analysis, all the strain can be calcu-lated explicitly. From the previous section it canbe noticed that both interaction strains are implicitfunctions of the total creep strain. Hence a predic-tor–corrector approach has been developed to eval-uate the interaction strains.

At a current time step i, the thermal and dimen-sional change strains are determined first. It isassumed that the average value in an incrementcan be approximately calculated by the central dif-ference method, i.e.

~r ¼ ri þDr2;

~e ¼ ei þDe2

.

ð35Þ


In the predictor step, all the strains except theinteraction strains are considered. Using Eqs. (13)and (35), the primary creep (19) and the secondarycreep (21) strains can be written as

Depc � uDcð2/1 þ /2Þ1þ 2uDc

Dee

¼ 2uDc/2eei þ 4uDcðeDcri � epc

i Þ1þ 2uDc

; ð36Þ

Desc � nDc2/1 þ /2

4

� �Dee

¼ nDc eDcri þ/2e

ei

2

� �; ð37Þ

where /1 = E/Ec is the ratio of Young’s modulus tocreep modulus and /2 = DE/Ec is the ratio ofchange in Young’s modulus to creep modulus. Thetotal strain in Eq. (11) can be written as

Dee þ Depc þ Desc ¼ Detotal � Dedc � Deth. ð38ÞNow we have three Eqs. (36)–(38) with known right-hand sides and three unknowns Dee, Depc and Desc,which can be solved easily using linear algebra. Thesolution for Dee is

Dee ¼Detotal � Dedc � Deth � nDceDcri �

4u DcðeDcri�epciÞ

1þ2u Dc � /2 Dc 2u1þ2u Dcþ

n2

� �ee

i

1þ Dc u1þ2u Dcþ

n4

� �ð2/1 þ /2Þ

. ð39Þ

Once the solution for Dee is found, the solutions forDepc and Desc can be found by substituting Dee intoEqs. (36) and (37).

Once we have the solutions for Depc and Desc,accordingly the total creep strain can be foundand hence both interaction strains. In the correctorstep, we update Eq. (38) as

DeeþDepcþDesc¼Detotal�Dedc�Deth�Deidc�Deith.

ð40ÞA new approximation for Dee can be obtained byreplacing Eq. (38) with Eq. (40) in the predictorstep, and hence the new approximation can befound for the creep strains and the interactionstrains.

In theory, improved approximations to Dee canbe obtained by iterating the corrector step. In prac-tice, since Dee converges to the actual approxima-tion given by the implicit formula rather than tothe solution of Dee, it is more efficient to use a reduc-tion in the step size if improved accuracy is needed.

6. Numerical example

As an example, a simple three-dimensional modelhas been analysed as depicted in Fig. 4. The modelrepresents one-eighth of a half-height graphite mod-erator brick used in a hypothetical reactor design.The internal radius ri and external radius re are150 mm and 300 mm, respectively. The modelheight is 250 mm. Fig. 5 shows the dimension ofthe model cross-section. The radius of the cornerat the root of the keyway is 6 mm. Symmetry condi-tion are applied on both radial boundaries and onthe top surface.

In order to perform the stress analysis, four fieldvariables are required. They are irradiation fluence,irradiation temperature, weight loss and radiolyticoxidation rate. Simple profiles for these field vari-ables have been created to model the reactor condi-tions. It is assumed that all field variables vary inradial direction but are uniform along the circum-ferential direction and the height of the brick.Fig. 6 shows the irradiation fluence profile at theend of 30 full power years (fpy). It is assumed thatthe maximum and minimum values for these field

variables occur at the internal radius and externalradius, respectively. Both irradiation fluence andweight loss are assumed to vary linearly with timewhereas the temperature and the oxidation rateare assumed to remain constant through the wholereactor life. Before reactor start-up and after shut-down the brick is assumed to be at a uniform ambi-ent temperature. The material properties used in theanalysis are taken from Brocklehurst and Kelly [27].

The undeformed shape of the brick is shown inFig. 7 together with the irradiated brick shape after30 full power years. The graphite moderator brickchanges its shape during its service life and the over-all dimensions become smaller; both the height andthe bore radius of the brick decrease by about 3%.When fuel bricks are shrinking the hoop stressesare tensile at the bore and compressive at theperiphery, leading to keyway closure and keywaydovetailing. After the turnaround as the graphiteswells the situation is reversed and the keyways startto open.

Fig. 4. Finite element mesh profile of a hypothetical graphitemoderator brick; the length of the model is 250 mm, symmetrycondition is applied on both radial boundaries and on the topsurface.

Fig. 5. Dimensions of the graphite brick cross-section.

Fig. 6. Irradiation fluence profile at the end of 30 fpy, maximumfluence at the inner radius and minimum fluence at the outerradius.

Fig. 7. Comparison of an unirradiated brick (dotted line) with anirradiated brick (solid line) after 30 fpy.


The bore hoop stresses at three height positionsare shown in Fig. 8. Initially, at start-up compres-sive stresses are caused by the radial temperature

variation; the temperature being the highest at thebore. These thermal stresses are quickly relievedby the primary creep strain. As the irradiationinduced shrinkage of the graphite is initially highest

Fig. 8. Time history of the bore hoop stress at three heights.


in the high temperature region near the bore, thestresses quickly change sign with increasing fluenceand tensile stresses are developed at the bore.Towards the end of the brick life, compressive stres-ses develop again at the bore due to expansion ofthe graphite after turnaround. The crossover instress occurs around 22 fpy. It is important to notethat the shut-down stresses are higher than the oper-ating stresses. This is because the initial thermalstresses are crept out during operation, but returnin the opposite sense at shut-down and add to theshrinkage stresses, in addition the irradiationchange in CTE also increases the shut-downstresses.

The maximum tensile hoop stresses at z = 0 mm,125 mm and 250 mm are 1.41 MPa, 6.65 MPa and7.66 MPa, respectively. It is clear that the stressesvary with height along the brick length. The stresslevel is highest at z = 250 mm (i.e. middle of thebrick). The stress level is lowest at z = 0 mm (i.e.bottom of the brick).

The stress profiles at the keyway are shown inFig. 9 and are in opposite in sign to those in the

Fig. 9. Time history of the keyway hoop stress at three heights.

bore. The initial keyway stress is tensile at the key-way. As irradiation damage began the stresseschange sign and compressive stresses develop. Thestresses change sign again due to dimensionalchange turnaround.

The maximum compressive hoop stresses atz = 0 mm, 125 mm and 250 mm are 14.95 MPa,7.98 MPa and 6.43 MPa respectively. Hence, themaximum compressive hoop stress now occurs atthe bottom of the brick.

7. Conclusions

There are significant property changes in nucleargraphite components due to fast neutron irradiationand radiolytic oxidation. These changes lead tocomponent deformations and stresses that must beaccounted for in graphite component life timeassessments. A complex material model for nucleargraphite under irradiation and radiolytic oxidationconditions has been derived. The model is largelyempirical and is based on measurements made ongraphite samples irradiated in material test reactors.A predictor–corrector procedure has been devel-oped to solve the material model constitutive equa-tions. The model has been implemented in, and canbe used together with finite element program ABA-QUS to perform irradiated graphite stress analyses.A stress analysis is presented for a three-dimen-sional hypothetical nuclear graphite brick. Thenumerical results predict that the maximum tensilestress at the bore occurs at the middle height ofthe brick, whereas the maximum compressive stressat the keyway occurs at the bottom of the brick.

References

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[10] S. Mrozowski, Mechanical strength, thermal expansion andstructure of cokes and carbons, in: Proceedings of the 1stConference on Carbon, 1956, p. 31.

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[18] D.K.L. Tsang, B.J. Marsden, Modelling of nuclear graphitebehaviour under irradiation conditions, in: Proceeding ofAgeing Management of Graphite Reactor Cores, Cardiff,2005.

[19] B.T. Kelly, T.D. Burchell, Carbon 32 (1994) 119.[20] T. Oku, K. Fujisaki, M. Eto, J. Nucl. Mater. 152 (1988) 225.[21] D.K.L. Tsang, B.J. Marsden, S.L. Fok, G. Hall, Carbon 43

(2005) 2902.[22] B.T. Kelly, Carbon 30 (1992) 379.[23] J.E. Brocklehurst, B.T. Kelly, Carbon 31 (1993) 179.[24] G.B. Neighbour, J. Phys. D: Appl. Phys. 33 (2000) 2966.[25] B.T. Kelly, Carbon 20 (1982) 3.[26] B.T. Kelly, T.D. Burchell, Carbon 32 (1994) 499.[27] J.E. Brocklehurst, B.T. Kelly, Carbon 31 (1993) 155.

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