The relationship between irradiation induced dimensional change and the coefficient of thermal...

Nuclear Engineering and Design 222 (2003) 319–330

The relationship between irradiation induced dimensionalchange and the coefficient of thermal expansion:

a modified Simmons relationship

G. Hall∗, B.J. Marsden, S.L. Fok, J. SmartNuclear Graphite Research Group, University of Manchester, Oxford Road, Manchester M13 9PL, UK

Received 14 March 2002; received in revised form 19 December 2002; accepted 20 December 2002

Abstract

In the 1960s, a theoretical relationship between the dimensional changes and the coefficient of thermal expansion of irradiatedgraphite was derived by J.H.W. Simmons. The theory was shown to be comparable with experimental observations at lowirradiation doses, but shown to diverge at higher irradiation doses. However, various modified versions of this theory have beenused as the foundation of design and life prediction calculations for graphite-moderated reactors.

This paper re-examines the Simmons relationship, summarising its derivation and assumptions. The relationship was thenmodified to incorporate the high dose, high strain changes that were assumed to be represented in the changes in Young’smodulus with irradiation dose. By scrutinising the behaviour of finite element analyses, it was possible to use a modifiedSimmons relationship to predict the dimensional changes of an isotropic and anisotropic graphite to high irradiation doses.

These issues are important to present high-temperature reactors (HTRs) as the life of HTR graphite components is dependentupon their dimensional change behaviour. A greater understanding of this behaviour will help in the selection and developmentof graphite materials.© 2003 Elsevier Science B.V. All rights reserved.

1. Introduction

Graphite is used as a neutron moderator in manynuclear reactors around the world, and because themoderator cannot be replaced in most of these reac-tors, the working life of the reactor maybe governedby its life. When subjected to fast neutron irradiation,the dimensions and properties of the graphite changeconsiderably. Three of the most significant are thedimensional changes, the coefficient of thermal ex-pansion (CTE) and the Young’s modulus. During the

∗ Corresponding author. Tel.:+44-161-275-4391;fax: +44-161-275-4346.

E-mail address:[email protected] (G. Hall).

design life of a reactor, the combined effect of thechanges is often difficult to predict accurately; the sit-uation is further complicated in carbon dioxide cooledreactors by corrosion due to radiolytic oxidation.

The relationship between changes in CTE and di-mensions in irradiated graphite has been investigatedby a number of authors. One notable set of rela-tionships is those derived bySimmons (1961)fordimensional change rate and coefficient of thermalexpansion. These relationships associate the macro-scopic properties of a polycrystalline graphite to theproperties of graphitic crystallites. At low doses thedimensional change relationship shows good agree-ment with experimental data. However, at higherdoses, theory and experimental data diverge as the

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320 G. Hall et al. / Nuclear Engineering and Design 222 (2003) 319–330

relationship does not predict the reversal of shrinkage(turnaround) exhibited in many graphites.

This paper re-examines the derivation of the Sim-mons relationships and presents a modification tothem so that turnaround could be demonstrated. Thepredictions are then compared to experimental dataand results from a finite element method (FEM) studyalso being undertaken. A greater understanding ofthis relationship will aid the selection, design anddevelopment of new graphites.

2. Theory

2.1. Simmons relationship

Simmons (1961)hypothesised that dimensionalchanges in irradiated graphite are a result of changesto those properties in the graphite crystal lattice andchanges to the CTE. It had also been observed at lowdoses that there was a linear relationship betweendimensional change rate and CTE (Simmons, 1959).He concluded that a relationship between the CTEand dimensional changes with fast neutron irradiationcould be formed. Several assumptions were made:

(a) Polycrystalline graphite is a single-phase material,i.e. it is a porous aggregate of graphite crystals,with each crystal having the correct graphite crys-tal symmetry and each possessing identical prop-erties.

(b) Small elemental volumes (an) of polycrystallinegraphite may be chosen such that stresses inducedeither internally or externally can be regarded asuniform.

(c) The volume of each elementan contains onlygraphite and therefore, a unit volume of polycrys-talline graphite can be taken as being∑n

an = ρ

ρc(1)

whereρ is the bulk density andρc is the crystaldensity.

From the first law of thermodynamics, if a smallamount of heat (dQ) flows into a unit volume ofgraphite crystal and a small amount of work (dW) isdone, the increase in internal energy (dU) is given by

dU = dW + dQ (2)

The second law of thermodynamics states that fora reversible change

dQ = T dS (3)

whereS is the entropy andT is temperature. For smallstrains, the work done is given by

dW = σij dεij (4)

whereσij andεij are the tensor components of stressand strain, respectively (Nye, 1964). The free energy(F) of the crystal may now be written as

F = U − σijεij − TS (5)

All of these quantities are a function of temperature(T), stress (σij ) and irradiation dose (γ ), which definethe state of the unit volume of graphite crystal.

Considering a unit volume of graphite crystal, whichhas been irradiated at zero stress and at a fixed temper-ature by irradiation doseγ , the change in free energydue to the irradiation dose is

dF = dU − σij dεij − εij dσij − T dS − S dT (6)

CombiningEqs. (2)–(4) and (6)gives

dF = −εij dσij − S dT (7)

As the free energy is a function ofT, σij andγ , crystalstrain and crystal entropy can be defined as

εij = −(

∂F

∂σij

)T

, S = −(

∂F

∂T

)σ

(8)

Differentiating the first expression again with re-spect toT and the second expression with respectto σij , keepingT and γ constant and all except oneσij constant, the well-known expression for thermalexpansion is obtained.

αij =(

∂εij

∂T

)σ

=(

∂S

∂σij

)T

= − ∂2F

∂σij ∂T(9)

By differentiating the first expression inEq. (7)withrespect toγ , keepingσij andT constant, a similar ex-pression for irradiation growth or dimensional change(gij ) is obtained.

gij =(

∂εij

∂γ

)σ

= − ∂2F

∂γ ∂σij(10)

For thenth crystal,x and y are perpendicular to thec-axis (z direction) of the crystal. The stress in the

G. Hall et al. / Nuclear Engineering and Design 222 (2003) 319–330 321

crystal is denoted byσ ′rs,n, where the indicesr ands

are equivalent toi andj in the unit volume of graphite.Therefore, the relationship between bulk and crystalproperties can be shown to be

U(σij , T , γ ) =∑n

anU′(σ ′

rs,n, T , γ ),

S(σij , T , γ ) =∑n

anS′(σ ′

rs,n, T , γ ),

F (σij , T , γ ) =∑n

anF′(σ ′

rs,n, T , γ ) (11)

Thus, the dimensional change and CTE in the poly-crystalline graphite in some direction, say thex–xdirection, can be written as

dεxx

dγ= − ∂2F

∂γ ∂σxx= −

∑n

an

∑rs

∂2F ′

∂γ ∂σ ′rs,n

∂σ ′rs,n

∂σxx

αxx = dSxx

dσxx= − ∂2F

∂σxx∂T

= −∑n

an

∑rs

∂2F ′

∂σ ′rs,n∂T

∂σ ′rs,n

∂σxx(12)

To a first-order approximation,S′, E′ andF′ are evenfunctions of stress and therefore, only the normalcomponents of the right-hand side of this relationshipare non-zero. Thus,Eq. (12)becomes

dεxx

dγ=∑n

an

[∂σ ′

33,n

∂σxx

dxc

dγ+ ∂σ ′

22,n

∂σxx

dxa

dγ+ ∂σ ′

11,n

∂σxx

dxa

dγ

],

αxx =∑n

an

[∂σ ′

33,n

∂σxx

dSc

dσ ′33,n

+ ∂σ ′22,n

∂σxx

dSa

d∂σ ′22,n

+ ∂σ ′11,n

∂σxx

dSa

dσ ′11,n

](13)

where dxa /dg and dxc/dg are the crystal strains par-allel to the basal planes (a-axis) and perpendicular tothe basal planes (c-axis), respectively, and∂Sc/∂

′33,n,

∂Sa/∂′22,n and∂Sa/∂

′11,n are the crystal CTEs in the

c-axis and twoa-axis directions, respectively. There-fore, Eq. (13)can be rewritten as

dεxx

dγ=∑n

an

[∂σ ′

33,n

∂σxxgc + ∂

∂σxx(σ ′

22,n + σ ′11,n)ga

],

αxx =∑n

an

[∂σ ′

33,n

∂σxxαc + ∂

∂σxx(σ ′

22,n + σ ′11,n)αa

]

(14)

It can be shown (Cook and Young, 1999) that byconsidering three orthogonal directions in the bulkgraphite, the stress invariantJ1 is

J1 = σxx + σyy + σzz (15)

Partial differentiating this with respect to the stress inthe polycrystalline structure along thex–x directiongives

∂J1

∂σxx= ∂

∂σxx(σxx + σyy + σzz) = 1 (16)

Simmons (1961)states that the stress invariant canalso be defined by the summation of the crystallinestresses

J1 =∑n

an(σ′11,n + σ ′

22,n + σ ′33,n) (17)

Thus, partial differentiation with respect to the stressin thex–x direction gives

∂J1

∂σxx=∑n

an

(∂σ ′

11,n

∂σxx+ ∂σ ′

22,n

∂σxx+ ∂σ ′

33,n

∂σxx

)= 1

(18)

Letting

Ax =∑n

an

(∂σ ′

33,n

∂σxx

)(19)

(1 − Ax) =∑n

an

[∂

∂σxx(σ ′

22,n + σ ′11,n)

](20)

and relating the stress in the crystallites to the stressin the polycrystalline structure, combiningEqs. (14),(19) and (20), gives the final relationships

dεxx

dγ= Axgc + (1 − Ax)ga (21)

αxx = Axαc + (1 − Ax)αa (22)

where Ax is the structure factor. The CTEs of thecrystal (aa andac) and polycrystalline graphite (axx)must be defined over the same temperature ranges.


Therefore,Eq. (22)can be used to obtainAx as a func-tion of dose.

Ax = αxx − αa

αc − αa

(23)

However, as previously stated, Simmons’ relation-ships diverge at high dose. A number of other similarrelationships have been investigated by other authors(Brocklehurst and Kelly, 1993; Sutton and Howard,1962; Jenkins, 1964; Morgan, 1966).

2.2. Modified Simmons relationship

It was recently proposed (Hall et al., 2002) thatto account for the relatively high strain dimensionalchanges, an extra structure factor (Bx) should be in-troduced into the Simmons relationship. This extraterm was applied to the dimensional change formulagiving

dεxx

dγ= AxBxgc + (1 − AxBx) ga (24)

In irradiated graphite, the Young’s modulus hasbeen shown to be a function of irradiation dose andtemperature (Brocklehurst and Kelly, 1993). Thechange in Young’s modulus (Fig. 1), where E0 isthe unirradiated value andE, the irradiated value, isassumed to be a combination of two distinct mecha-nisms defined by

Fig. 1. Idealised components of the change in Young’s modulus with equivalent DIDO nickel (EDN) dose for an isotropic graphite at 450◦C.

E

E0=[

E

E0

]P

[E

E0

]S

= P · S (25)

The pinning term (P) accounts for the pinning of mo-bile dislocations in the graphite structure, and is as-sumed to act instantaneously and remain constant withirradiation dose but assumed to be a function of irra-diation temperature. The structure term (S) describesirradiation-dependent structural changes that vary withboth dose and irradiation temperature.

The authors (Hall et al., 2002) have already pre-sented the possibility of using the modified Simmonsrelationship and changes in the Young’s modulusstructure term to predict dimensional change. Theworking hypothesis was that for an isotropic poly-crystalline graphite,Bx could be defined by

Bx = (S − 1)a eS + 1 (26)

where S was the change in modulus structure anda was a constant. Although this gave a reasonableprediction of the dimensional changes at 450◦C, ithad not been applied to other temperatures and theequation was empirically based upon a fit to exper-imental data. Therefore, still using the suppositionthat the Young’s modulus and dimensional changeswere related, the relationship has been re-examined.The assumption in this paper is thatBx can be di-rectly related to the changes in Young’s modulusstructure term, i.e.Bx = S; the previous equation isunnecessary.


Fig. 2. Finite element method models.

2.3. Finite element method

A similar investigation (Marsden et al., 2001) intothe relationship between crystal and bulk graphiteproperties, namely dimensional, Young’s modulusand CTE changes with dose, has been conducted us-ing the FEM. Again, the underlying assumption wasthat the crystal behaviour was the dominant forcebehind the behaviour of polycrystalline graphite.

A model of an idealised crystal/filler particlesurrounded by binder material has been developed(Fig. 2). The filler model was subjected to loadingconditions akin to those in a reactor and its behaviour

Fig. 3. Comparison of the Simmons relationship with experimental dimensional changes at 430 and 600◦C for an isotropic graphite.

inserted into a polycrystalline model (Fig. 2). For anisotropic bulk graphite, the filler particles were ran-domly oriented and distributed in a matrix of bindermaterial and porosity; for an anisotropic bulk graphite,the filler particles would be preferentially aligned.This polycrystalline model was then subjected to thesame loading conditions as the filler particle model.The changes in the apparent property changes (di-mensional, CTE and Young’s modulus) were thenexamined and compared to those observed exper-imentally. The similarities in the behaviours couldthen be examined more closely in the FEM modelsand the microstructural change that is the cause of the


Fig. 4. Comparison of the modified Simmons relationship, dεxx/dγ = AxBxgc + (1− AxBx)ga , and experimental dimensional changes at430 and 600◦C for an isotropic graphite.

behaviour identified. This was then related to the realstructure and further evidence from experimental datasought. This would not only give an insight into whatwas happening to the structure but also the possibilityof predicting graphite behaviour more accurately.

3. Results

Simmons’s original and modified relationshipshave been used to calculate the dimensional changes

Fig. 5. Comparison of the FEM and experimental dimensional changes at∼430 and 600◦C for an isotropic graphite.

for an isotropic polycrystalline graphite. The crystal(highly oriented pyrolytic graphite) and polycrys-talline (Gilsocarbon) data at 430 and 600◦C wereobtained fromKelly (1971) and Brocklehurst andKelly (1993), respectively. Calculation of the dimen-sional changes at 430◦C using Simmon’s originalrelationship showed good agreement at low doses, butdiverged when the experimental data demonstratedturnaround at higher doses (Fig. 3). The same occursat 600◦C, with divergence occurring at a lower dose(Fig. 3).


Fig. 6. Changes in experimental and FEM Young’s modulus structure terms at∼430 and 600◦C for an isotropic graphite.

The modified Simmons relationship dεxx/dγ =AxBxgc+(1−AxBx)ga , with Bx equal to the changesin the Young’s modulus structure termS, predicteddimensional changes that did show turnaround at430◦C (Fig. 4). However, the amount of shrinkagewas less than the experimental data and as the doseincreased, the dimensional change rate began to re-duce, eventually shrinking again (>200 EDND). At600◦C, the behaviour demonstrated some degree ofturnaround but then it also began to contract again asthe dose increased (>130 EDND).

Fig. 7. Modification of the Young’s modulus structure term at 430◦C.

Examination of the FEM results showed that themodels at 450 and 600◦C predicted dimensionalchanges similar to those observed experimentally(Fig. 5). It should be noted that the FEM study onlywent to a maximum dose of 160 EDND and did notconsider 430◦C but 450◦C. However, the predictionsat 450◦C were very similar to the experimental dataat 430◦C. At 600◦C, the FEM prediction showedmore shrinkage than the experimental data, althoughthe dose at which turnaround occurred was similarin both. Neither temperatures exhibited the second


Fig. 8. Effect of modifying the Young’s modulus on the predicted dimensional changes at 430◦C.

shrinkage observed in the values calculated usingthe modified Simmons relationship. Comparison ofthe changes in Young’s modulus structure values forthese FEM models to those used in the modified Sim-mons relationship, the experimental trend changes inS, highlighted some important features (Fig. 6).

The experimental trends and FEM values showedsome similarities, although the actual values were dif-ferent. Both showed that the values at the higher tem-perature increased at a lower dose and at a higher rate,the gradient of the steady increase being similar forboth the experimental and FEM data at the same re-

Fig. 9. Modification of the Young’s modulus structure term at 600◦C.

spective temperature. The experimental data, however,showed an increase at a lower dose and to a highermagnitude when compared to the FEM results. The430◦C experimental data has a noticeable increase inS value at low doses whereas the others tend to re-main close to unity at these doses. The experimentaldata also exhibited a fall in Young’s modulus as thedose increased while the FEM shows only a reduc-tion in the rate of increase (although the FEM data at450◦C is probably not to a high enough dose). Relat-ing these behaviours and differences to the respectivedimensional changes raised a number of points.


Fig. 10. Effect of modifying the Young’s modulus on the predicted dimensional changes at 600◦C.

In the FEM models, there appeared to be an iden-tifiable relationship between the dimensional changeand changes in Young’s modulus. As the temperatureincreased the Young’s modulus increased at a lowerdose and at a faster rate. The respective dimensionalchanges showed that as the temperature increased,turnaround occurred at a lower dose and the dimen-sional change rates were higher. The same, however,cannot be said for the predicted data. The changes inYoung’s modulus may demonstrate a similar patternas the FEM results but the respective dimensionalchanges do not follow suit. The higher temperature

Fig. 11. Comparison of the modified Simmons relationship and experimental dimensional changes at 600◦C for an anisotropic graphite,parallel to the extrusion axis.

dimensional results had a higher dimensional changerate but turnaround at a higher dose than the lowertemperature values. However, the second shrinkageexperienced in the dimensional changes can nowbe attributed to the reduction in the experimentalYoung’s modulus at high doses. As the changes in theFEM results follow a logical pattern, could the exper-imental data, specifically theSvalues, be modified togive reasonable predictions?

If the data at 430◦C is examined first, it can be seenthat there was a significant amount of scatter withinthe data (Fig. 7). The trend used in the modified


Fig. 12. Comparison of the modified Simmons relationship and experimental dimensional changes at 600◦C for an anisotropic graphite,perpendicular to the extrusion axis.

Simmons’s relationship was calculated by averagingthe groups of values, both in the dose andS values.As it has already been shown that turnaround is pre-dicted at a lower dose, if theS curve shifts to theleft (temperature increase) turnaround can be delayedby shifting theS curve to the right (Figs. 7 and 8).Also, as the reduction in Young’s modulus at the highdoses caused the second shrinkage in the dimensionalresults, keeping the Young’s modulus increase at asteady rate prevented this (Figs. 7 and 8). The re-sultant prediction using the modified structure termcurve and modified Simmons relationship was thensimilar to the experimental observations.

Examining the dimensional data at 600◦C (Fig. 4)indicated that at this temperature the predicted di-mensional changes and turnaround were similar to theexperimental data. The most noticeable deviation wasthat the dimensional changes exhibited the secondshrinkage as the dose increased but, as in the 430◦Ccase, the dose at which the predicted turnaround oc-curred was lower than the experimental. Therefore,theScurve at 600◦C was shifted to the right and thecurve altered so that it also remained at a constantrate of increase (Fig. 9). The subsequent dimensionalchanges exhibited turnaround and then expansion(Fig. 10) similar to the experimental data.

Preliminary calculations, based upon the isotropicresults, have been conducted for an anisotropic poly-crystalline graphite (PGA) at 600◦C, the crystal data

and polycrystalline data again being fromKelly (1971)and Brocklehurst and Kelly (1993). Using the sameprocedure as the isotropic case, it was possible to pre-dict the dimensional changes in the parallel (Fig. 11)and perpendicular (Fig. 12) to extrusion directions thatwere similar to the experimental results.

4. Discussion

One of the main assumptions made by Simmonswas that polycrystalline graphite can be regarded asa single-phase material. However, nuclear graphite ismade from two constituents, filler and binder, whichare differentiable under the microscope. The argumentis that both of these are formed from the same graphitestructure and it is the crystallinity that differs. In thefiller particles, there is a higher degree of crystallinitythan the binder. Ifac andaa are the same for both, thedifference in structure can be ignored. The evidence atthat time suggested thatac did not vary significantlyandaa not at all with varying crystallinity (Simmons,1965). There is no new evidence to contradict thisand it is assumed to be still valid. The fact that therelationship holds at low doses, and at high doses whenmodified, also supports this.

At low doses, the dimensional changes predictedusing the original Simmons relationships were similarto experimental data for an isotropic polycrystalline


graphite at 430 and 600◦C (Fig. 3). There was a dif-ference at very low doses (<10 EDND) because of asmall amount of expansion observed at these doses.This is sometimes attributed to the stresses createdin the manufacturing process (Nightingale, 1962).The divergence of the results as the dose increasedindicated that at higher doses, another mechanism,not related to thermal expansion, has an effect on thebehaviour and must be considered.

The proposed modified relationship assumed thatthe high dose dimensional changes could be ac-counted for through the inclusion of an extra factorBx in the Simmons relationship. SettingBx to equalthe changes in Young’s modulus structure termS didnot immediately give reasonable predictions (Fig. 4).However, when the FEM results were compared to theexperimental data (Figs. 5 and 6), a number of pointscould be identified. The FEM dimensional changesand structure term changes showed similar trends atboth temperatures (∼430 and 600◦C) when comparedto the experimental data. The differences betweenthe FEM and experimental dimensional changes werematched by differences in the structure terms. A delayin turnaround when going from 600 to 430◦C waslinked to a delay in increasingS. In the FEM study,this delay has been attributed to there being less ac-commodation at the higher temperature as the crackswithin the structure have closed further due to thehigher temperature. Therefore, the complete closureof the cracks occurred at a lower dose at the highertemperature and thus, turnaround was earlier at thehigher temperature. This may also be true in the realstructure. The differences in the dimensional changerates (600◦C was higher than 430◦C) was shown inboth sets of data. A similar difference was also shownin the structure term data. In the 600◦C FEM model,the dimensional change rate was higher because thecrystal dimensional change rate was higher at 600◦Cthan 430◦C. Thus, the same may be true for thereal structure. In the experimental data, there was areduction in the structure termS at high doses. Thisreduction was therefore seen in the predicted dimen-sional changes in the form of a second shrinkage. Inthe FEM data there was no reduction in value, just areduction in the rate of change. A likely cause of thereduction inSwas cracking or fracture, which was notincluded in the FEM models. Thus, the FEM dimen-sional changes continued to expand after turnaround

and did not show the second shrinkage. Therefore,the experimental values could be used to calculate thedimensional changes, but only after modification.

A shift in theScurve was assumed to accelerate ordelay the onset of turnaround. In the 430◦C results,turnaround occurred at a lower dose than the experi-mental data. To compensate for this, theS curve wasshifted to the right (Fig. 7) and there was a subsequentdelay in turnaround (Fig. 8). As the FEMScurves didnot show a drop at high doses, the experimental trendat 430◦C was extended so that the increase in changeremained constant. The resulting dimensional changewhen this modified curve was used was closer tothe experimental data.

The data for the 600◦C curves were considered inthe same way (Figs. 9 and 10) except that the shiftof the 600◦C S-curve was much less than that in the430◦C case. The reason for this difference is unknownat this time but it could be a result of inaccuracies ofthe experimental data used, inaccuracies in the calcu-lations and assumptions, or another mechanism that isdependent upon temperature needs to be considered.The rate of dimensional change after turnaround in thepredicted results appeared to be less than the experi-mental observations. In order to examine this further itwould be necessary to obtain more data up to a higherdose.

The modified relationship was also used to predictthe dimensional changes in an anisotropic graphitecase. Again, theS curves (parallel and perpendicu-lar) required modification. The resulting predictions(Figs. 11 and 12) were close to the experimental data,giving more confidence in the procedure.

This investigation has reiterated that at low dosesthe original Simmons relationship holds, but as doseincreases, another mechanism needs to be included.By using a modified Simmons relationship and thechange in Young’s modulus structure term, it is pos-sible to predict with reasonable accuracy, the dimen-sional changes of an isotropic graphite at 430 and600◦C, and an anisotropic graphite at 600◦C. How-ever, the Young’s modulus structure term curve hadto be modified to give reasonable dimensional changepredictions at the very high doses. This indicates thatthere is either another mechanism to consider or thatthe mechanism to be used in the relationship is not infact the change in Young’s modulus structure term, butsome mechanism that affects the changes in Young’s


modulus. One possibility is that the change in inter-nal stress is the mechanism that should be used. Theproblem with this is that it would be difficult to obtainexperimental data on this parameter, and thus, diffi-cult to make predictions using the relationship. Theadvantage of the changes in Young’s modulus is thatin comparison to other properties, it is relatively easyto measure.

In order to be more confident of this modified rela-tionship, it would be necessary to investigate a numberof areas. More data from other sources should be col-lected at the same temperatures (430 and 600◦C) andthe relationship applied. Data at other temperaturesshould also be considered as there can be differences inbehaviour as the temperature changes. If the relation-ship is correct, it should be tested with other graphitetypes, isotropic and anisotropic. The perceived prob-lem with all of these is obtaining the experimental datafor the crystal (aa , ac, ga , gc) and polycrystalline (axx ,S) graphites at the required temperatures and over asufficient dose range. It would also be advantageousto compare FEM models of these graphites with theobserved and predicted behaviours.

5. Conclusions

(a) Simmons’ relationship and its derivation havebeen re-examined and the assumptions stated.

(b) The Simmons relationship between irradiation in-duced dimensional changes and CTE have beenshown to give close approximations of isotropicgraphite (430 and 600◦C) and anisotropic graphite(600◦C) at low irradiation doses.

(c) Because the original Simmons relationship brokedown at high doses, a modified Simmons relation-ship that incorporates high dose, high strain mech-anisms, has been proposed.

(d) By relating the high dose dimensional changemechanisms to the changes in Young’s moduluswith irradiation dose, good predictions of isotropic(430 and 600◦C) and anisotropic (600◦C) di-mensional changes have been obtained. The re-lationship between modulus structure term anddimensional change through the Simmons struc-ture factor is direct until the volumetric changeapproaches zero at high dose, after which thedirect relationship breaks down.

(e) Finite element method models have been used tocompare and idealised graphite microstructure andits behaviour to observed and predicted results.From these it was possible to obtain some insightinto the mechanism behind the behaviours.

Acknowledgements

This work was funded by British Energy plc whileconducting an EngD at the University of Manchester.

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