Journal of Nuclear Materials - Inside Mines

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Journal of Nuclear Materials 511 (2018) 91e108

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Journal of Nuclear Materials

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Characterisation of the spatial variability of material properties ofGilsocarbon and NBG-18 using random fields*

Jos�e David Arregui-Mena a, *, Philip D. Edmondson a, Lee Margetts b, D.V. Griffiths c,William E. Windes d, Mark Carroll d, Paul M. Mummery b

a Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USAb University of Manchester, Oxford Road, Manchester, M13 9PL, UKc Colorado School of Mines, 1500 Illinois St, Golden, CO, 80401, USAd Idaho National Laboratory, Idaho Falls, ID, 83415, USA

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:Received 31 March 2018Received in revised form6 September 2018Accepted 6 September 2018Available online 8 September 2018

* This manuscript has been authored by UT-Battellethe publisher, by accepting the article for publicationreproduce the published form of this manuscript, or alresearch in accordance with the DOE Public Access P* Corresponding author. Tel.: þ1 865 241 3874.

E-mail addresses: [email protected], daviditi

https://doi.org/10.1016/j.jnucmat.2018.09.0080022-3115/Published by Elsevier B.V.

a b s t r a c t

Graphite is a candidate material for Generation IV concepts and is used as a moderator in Advanced Gas-cooled Reactors (AGR) in the UK. Spatial material variability is present within billets causing differentmaterial property values between different components. Variations in material properties and irradiationeffects can produce stress concentrations and diverse mechanical responses in a nuclear reactor graphitecore. In order to characterise the material variability, geostatistical techniques called variography andrandom field theory were adapted for studying the density and Young's modulus of a billet of Gilso-carbon and NBG-18 graphite grades. Variography is a technique for estimating the distance over whichmaterial property values have significant spatial correlation, known as the scale of fluctuation or spatialcorrelation length. The paper uses random field theory to create models that mimic the original spatialand statistical distributions of the original data set. This study found different values of correlation lengthfor density and Young's modulus around the edges of a Gilsocarbon billet, while in the case of NBG-18,similar correlation lengths where found across the billet. Examples of several random fields are given toreproduce the spatial patterns and values found in the original data.

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J.D. Arregui-Mena et al. / Journal of Nuclear Materials 511 (2018) 91e10892

1. Introduction

Graphite components are deployed in multiple present-daynuclear power stations including the Advanced Gas-cooled Re-actors (AGR) in the UK and will be part of Generation IV designssuch as the Very High Temperature Reactor (VHTR) and someMolten Salt Reactors (MSR) concepts. The graphite core of thesereactors designs serve as a moderator of fast neutrons, repositoryfor fuel, and to provide structural support [1]. The reactor envi-ronment promotes several ageing mechanisms in the graphitecomponents that lead to material property and dimensionalchanges. Examples of these mechanisms are the irradiationdimensional changes, irradiation creep, thermal strains andoxidation. The combination of these ageing and degradationmechanisms may lead to the distortion and internal stresses ofindividual components that in time can result in the formationand propagation of cracks. When enough graphite componentscrack or deform, the geometry of the core may interrupt thenormal operations of a nuclear reactor, such as refuelling oper-ations and the cooling thermodynamics of the graphite core.Thousands of graphitic bricks form the reactor core, the me-chanical response of each of these components is highlydependent on the fluence, temperature profile, coolant and me-chanical properties of the specific graphite grade. A certain de-gree of mechanical property variability is expected to be foundinside and in-between billets. This variability is normally notaccounted by the assessment and inspections on the graphitecore.

Part of the routine assessments and predictions of the lifetimeof graphite components include Finite Element Method (FEM)simulations. Assessments based on FEM models require anextensive database of the unirradiated and irradiated response ofgraphite. Some examples of FEM analysis for AGR componentscan be found in literature [2e4]. In general, these models includethe effects of thermal expansion, irradiation-induced dimen-sional changes, irradiation creep and other ageing mechanismsthat may alter the constitution of a graphite component. Thestructural integrity studies of graphite components are usuallysupported by stress analysis and other calculations that estimatethe failure rate and lifetime of the components. In addition tostress analysis, several fracture mechanics techniques have beenimplemented to analyse the crack evolution in nuclear graphitecomponents [5e9]. Traditionally FEM and continuum damagetechniques use the mean values of material properties to performtheir analyses. Unfortunately, by only using the mean value of thematerial properties all the spatial and statistical information suchas standard deviation and probability distribution are lost. Aprevious study that includes spatial material variability into FEMhas demonstrated that heterogeneity of material properties inAGR reactors can lead to stress concentrations [10].

Quantifying the degree of heterogeneity of graphite compo-nents is essential to predict possible differences on the failurerate of graphite components. Inconsistencies in material prop-erties may lead to the generation of stress concentrations e thiseffect can be produced by the combination of a temperaturegradient and a significant heterogeneity of material properties inthe graphite bricks [11]. A similar effect can be expected to beproduced by the combination of irradiation-induced dimensionalchange and material property variation within a single graphitecomponent.

The objectives of this research are to characterise the material'sspatial variability through geostatistical techniques and reproducethe spatial fluctuations of two grades of nuclear graphite throughmathematical models called random fields. The characterisation ofspatial fluctuations only requires additional calculations that

provide a new insight on the variations of materials propertieswithin a billet. Alternatively, random field realisations reproducemodels of the spatial fluctuations found in graphite allowing themodeller and designers to simulate different scenarios. Thisresearch focuses on the density and dynamic Young's modulusparameters of Gilsocarbon and NBG-18, although other materialproperties of interest can also be studied using this methodology.

1.1. Background

Several studies have shown the presence of heterogeneity ofmaterial properties in graphite components. These studies havebeen carried out for Gilsocarbon, the moderator of AGRs [12e14],H-451 used at Peach Bottom and Ft. St. Vrain reactors (USA) [15,16]and NBG-18 [17] a modern graphite grade proposed for futureVHTRs.

Gilsocarbon is a medium grain graphite composed of sphericalonion-shaped filler particles that are moulded during themanufacturing process [18]. The combination of moulding andspherical particles in Gilsocarbon result in an isotropic or semi-isotropic mechanical behaviour. The name of Gilsocarbon gradewas given to the family of graphite grades designed for the AGRs inthe UK. All these grades differ from each other as they were man-ufactured by two different companies, Anglo Great Lakes Corpo-ration Limited (AGL) and Union Carbide. Although all these gradeswhere manufactured in similar conditions, all have slightlydifferent grain sizes and microstructure. Several studies wereconducted on the possible variations of properties among thesegraphite grades and within single billets. The first published mea-surements on the spatial variability of the physical and mechanicalproperties in Gilsocarbon within a single billet were conducted byPreston [13,14]. Approximately half of a billet was sectioned tomeasure several types of physical and mechanical properties. Thesemeasurements include density, electrical resistivity, coefficient ofthermal expansion, four-point bending strength, compressivestrength, Young's modulus, tensile strength, thermal conductivity,open pore volume, closed pore volume and Poisson's ratio. Thesestudies confirmed the spatial variability of material's physical andmechanical properties. However, the data by Preston are not idealfor calibrating a random field for the purposes of this study.Therefore, another source of density and dynamic Young's moduluswas used for the calculations of this study [19].

The second material examined here, NBG-18, was a candidategrade for a component of the Pebble Bed Modular Reactor (PBMR)and is also being proposed as a material for other types of VHTRs.This graphite grade is manufactured from pitch coke and vibra-tionally moulded with a medium grain size (about 1.6mm)[20,21]. An extensive qualification research program was devel-oped for this grade at Idaho National Laboratory. Part of thisprogram focused on the characterisation of a single unirradiatedbillet of NBG-18 and is also a guideline for future studies for othergrades of graphite. For this research a billet of this grade wassectioned into 770 specimens that were used to measure thedensity, compressive strength, tensile strength, Young's modulus,shear modulus and other mechanical properties. The mechanicalperformance of this graphite grade under irradiation is alsocurrently being investigated under the Advanced Graphite Creepprograms (AGC) AGC-1 [21] and AGC-2 [22].

2. Materials

2.1. Gilsocarbon data

The density and Young's modulus variability of material prop-erties were obtained by sectioning a billet of Gilsocarbon provided

Fig. 1. Gilsocarbon components: a) Tested Gilsocarbon billet, b) Example of a Hartlepool/Heysham AGR channel brick.

Fig. 2. Cutting plan for the Gilsocarbon graphite billet: a) Upper view, b) Isometric view.

Fig. 3. Density values for Gilsocarbon specimens.

J.D. Arregui-Mena et al. / Journal of Nuclear Materials 511 (2018) 91e108 93

Fig. 4. Dynamic Young's modulus values for Gilsocarbon specimens.


by EDF Energy Nuclear Generation Ltd. This billet was manufac-tured by Union Carbide for the Torness/Heysham II reactors (heat4904; serial number 07V3632). The samples were sectioned from abillet with dimensions of 930mm in height and 470mm in diam-eter (Fig. 1a). The geometry and dimensions of an AGR reactor canvary depending on the nuclear power station. A typical channelbrick diameter size is approximately 460mm and has a heightbetween 850 and 900mm. Fig. 1b shows a channel brick of Har-tlepool/Heysham AGR bricks.

The billet was cut into 508 cubic samples to obtain the densityand dynamic Young's modulus of the samples. ASTM standardswere followed to conduct the measurements: ASTM StandardC559-90 [23] for density and ASTM standard C769-09 [24] for dy-namic Young's modulus (The complete data set can be found inRef. [25].). A subset of 328 samples were used for the calculations ofthis research and the locations of these samples are shown in Fig. 2.The regions where the samples were obtained from are labelled asspines. The position and values of density and dynamic Young'smodulus for the spine sections can be found in Fig. 3 and Fig. 4,respectively. For a more detailed description of the acquisition anddata information the reader is referred to Reference [25].

The summary of the density and Young's modulus values forGilsocarbon are divided in Table 1 and Table 2. These statisticalvalues were used as input values for the random field generatordescribed in the results sections.

2.2. NBG-18 data

The measurements for the NBG-18 data were obtained from acommercial block of graphite labelled NBG-18 635-14. The originaldimensions of the tested billet were1828.8mm� 609.6mm� 508mm. This block was sectioned intosmaller samples to produce samples suitable for density, dynamicYoung's modulus, Coefficient of Thermal Expansion (CTE) andstrength tests. Example prismatic block designs for the VHTR isshown in Fig. 5a (fuel block) and Fig. 5b (reflector block). Theapproximate dimensions of this design of bricks are 787.4mm talland 406.4mm diameter. The data points locations and values ofdensity and Young's modulus for NBG-18 are represented in thedata maps of Fig. 6 and Fig. 7. A more in-depth description of themeasurements and data can be found in Ref. [17]. Density and dy-namic Young's modulus measurements were obtained followingthe ASTM C559 [23] and ASTM C769 [24], respectively.

The Young's modulus of this data set was measured in two di-rections, against-grain andwith-grain orientations. For the purposeof this study the data for both directions were combined as one. Inorder to obtain the desired calculations, the data points weregrouped in 4 regions to produce two-dimensional data sets. Thecentroids of all the samples that are within the same range arealigned in the x-direction; the 4 regions for each group are shownin Fig. 8. An example of how the data points are aligned for Region 1are shown in Fig. 8c and d. Table 3 and Table 4 summarise thedensity and Young's modulus for all the data points and individualregions. These values were used to produce the random fields forNBG-18.

3. Methods

In order to improve the understanding of spatial material vari-ability within a data set it is necessary to create and interpret avariogram. A variogram estimates the shape, correlation length orrange, and direction of spatial autocorrelation. The process to createa variogram can be divided into two steps: (1) the first is tocalculate an experimental variogram and (2) the second step is toderive a variogram estimator from the experimental variogram.

These two steps are known collectively as variography. A compre-hensive description of the methodology to generate a variogramand aspects of it can be found in Ref. [26]. Then, experimentalvariogram parameters and descriptive statistics of the spatial dataare processed to generate random fields for graphite components.

Table 1Mean and standard deviation for density of Gilsocarbon data.

Section Density mean value (kg/m3) Density standard deviation (kg/m3) Number of samples

All spines data 1843.34 12.19 328Spine 3 1830.53 6.48 80Spine 6 1838.02 8.17 80Spine 9 1853.07 6.73 84Spine 12 1851.80 8.66 84

Table 2Mean and standard deviation for Young's modulus of Gilsocarbon data.

Section YM mean Xdirection (GPa)

YM standard deviation Xdirection (GPa)

YM mean Ydirection (GPa)

YM standard deviation Ydirection (GPa)

YM mean Zdirection (GPa)

YM standard deviation Zdirection (GPa)

Number ofsamples

All spinesdata

12.945 0.445 12.742 0.468 12.911 0.439 328

Spine 3 12.381 0.182 12.192 0.292 12.446 0.254 80Spine 6 12.869 0.349 12.669 0.371 12.704 0.282 80Spine 9 13.290 0.186 13.005 0.224 13.157 0.207 84Spine 12 13.244 0.247 13.108 0.285 13.334 0.275 84

YM e Young's modulus.


3.1. Experimental variogram

The methodology and Matlab software used in this research iscontained in the manuscript by Trauth [27]. All calculations wereperformed using Matlab version R2016a. The general procedure tocalculate the experimental variogram is explained in the followingsteps and in Fig. 9:

a) Pair data points. Coordinates and data points for each value ofdensity or Young's modulus are stored in the form of a vector tocreate 2-D grid coordinates. Then each of these vectors areprocessed with the Matlab function meshgrid to store the in-formation and pair the data points for the subsequent calcula-tions. A total of 6 matrices are created and called X1, X2, Y1, Y2,Z1, Z2, where X1 are the coordinates in x direction Y1 are thecoordinates in y direction, Z1 are the observations of density orYoung's modulus and X2, Y2 and Z2 are the transpose matricesof X1, Y1 and Z1, respectively.

b) Separation distances. Separation distances of each paired datapoints are calculated with Equation (1) and stored in a matrixcalled DT. The individual distances in D or separation vectorsbetween two points are called lag.

DT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðX1 þ X2Þ2 þ ðY1 þ Y2Þ2

q(1)

c) Semivariances. Individual semivariances are estimated for eachpaired data points with Equation (2), h represents the lag.

gðhÞ ¼ 12ðZ1 � Z2Þ2 (2)

Fig. 5. Hexagonal VHTR. a) Fuel block, b) Reflector block.

d) Initial lag. A minimum distance or lag is determined be-tween points for posterior calculations due to the existenceof an uneven grid. This minimum distance is called initial lagand is computed with Equation (3).

lagmin ¼ minðDÞn

(3)

e) Maximum distance and maximum number of lags. Amaximum distance limit is calculated from the D matrix(Equation (4)). A good initial estimator of the maximum dis-tance is half of the distance of the largest distance contained inthe D matrix. Another important constrain in the calculations isthe maximum number of lags, this number is calculated byEquation (5). The maximum number of lags can be obtained bydividing the maximum distance calculated over the initial lag,this value is rounded to obtain an integer number. Both pa-rameters are used to maintain the stability of the variogram andto reduce the number of calculations.

Max distance ¼ maxðDÞ2

(4)

Max lagszMax distance

lagmin(5)

f) Classification of separation distances. An iterative processclassifies and calculates the values of variogram estimator.

g) Variogram. The last step consists in plotting the variogram es-timators versus the mean separation distances.

Fig. 6. Density map for NBG-18.


3.2. Variogram estimator

The experimental variogram plot contains some parameters andfeatures that are important to address. The first important feature isthat the variance increases as the lag distance departs from theorigin. Variogram models share important parameters that includethe nugget effect, the correlation length or range and sill, and thesevariogram characteristics are shown in Fig. 9g.

Nugget effect (n): Ideally this value should be zero (n¼ 0),although some factors can displace the origin of the variogramcurve. The nugget effect can also be interpreted as an abrupt changeof value at a small distance. This abrupt change is sometimes

Fig. 7. Dynamic Young's mo

present in some of the variograms shown in this research.Correlation length/range (L): In context of the variogram this

parameter represents the distance in which the variogram reachesa plateau. The range or correlation length is also the distance inwhich data points larger than this value become spatially uncor-related. The correlation length can also be interpreted as a mea-surement that determines the likelihood of finding similar values ina certain range. This behaviour is caused by the fact that points thatare paired at larger distances tend to have a lower autocorrelationbetween them. In other words a high correlation lengthmeans highhomogeneity through the material. Examples on the effect of thecorrelation length or range on distribution or variation in property

dulus map for NBG-18.

Fig. 8. Regions for NBG-18 data. a) x-y View of the regions, b) x-z View of the regions, c) Example of the alignment of data points in the x direction, x-y view for region 1, d) Exampleof the alignment of data points in the x direction, x-z view for region 1.

Table 3Mean and standard deviation for density of NBG-18 data.

Section Density mean value (kg/m3) Density standard deviation (kg/m3) Number of samples

All regions data 1848.56 8.59 517Region 1 1842.20 9.67 119Region 2 1850.81 5.93 141Region 3 1850.53 6.73 137Region 4 1849.92 9.06 119

Table 4Mean and standard deviation for Young's modulus of NBG-18 data.

Section YM mean direction (GPa) YM standard deviation (GPa) Number of samples

All regions data 12.106 0.264 253Region 1 12.076 0.259 63Region 2 12.156 0.250 65Region 3 12.116 0.256 71Region 4 12.068 0.296 54

YM e Young's modulus.


are distributed among 3 realisations of random fields with differentcorrelation length values (Fig. 10). The input values to create theserealisations were the mean, standard deviation and correlationlength. The values chosen for the average and standard deviation

came from the density data of Gilsocarbon, and these values are1834.34 kg/m3 and 12.19 kg/m3, respectively.

Fig. 10 illustrates the changes produced by different ranges orcorrelation lengths. As shown in Fig. 10a, the erratic distribution

Fig. 9. Methodology used to calculate the experimental variogram.

Fig. 10. These cubes have a side length of 100mm, mean density of 1834.34 kg/m3 and standard deviation of 12.19 kg/m3. a) Correlation length of 1mm, b) Correlation length of10mm, c) Correlation length of 100mm.


of values of density created by a low correlation length, the lowvalue of correlation length with respect of the cube (100mm)create “islands” or spots where the density values are highlycorrelated. In the case shown in Fig. 10b it is possible to observelarger patches with similar density values. The last case, Fig. 10cpresents more smooth transitions of density values throughlarger distances caused by a higher correlation length value. It isimportant to mention that a single realisation would not berepresentative of the possible outcomes of a random fieldgenerator.

Sill (C): This parameter is the maximum value that the vario-gram takes and where the variogram reaches a plateau or tend tobecome linear. This value is usually close to the variance of the data[28].

A Matlab code by Schwanghart [29] was used to fit a function to

the experimental variogram obtained in the last subsection. Thiscode uses a function called variogramfit that requires the variogram

Fig. 11. Top-down approach, local average subdivision method process.

Fig. 12. Input values for the Local Average Subdivision (LAS) Method.


calculations, desired variogram models and parameters. The codeby Schwanghart uses a least squares fit to obtain the theoreticalvariogram from the experimental variogram.

The chosen variogram model for this research was a negativeexponential model; the function that describes this model

Fig. 13. Variograms for Gilsocarbon density data. a) Spine 3 variogram,

approaches the sill asymptotically. Because of this, the model doesnot have a practical finite range; however, a practical or effectiverange is assumed to be equal to 95% of the sill variance. Theexponential model presents a linear behaviour near its origin; thislinearity can be used to estimate the sill by plotting a straight line

b) Spine 6 variogram, c) Spine 9 variogram, d) Spine 12 variogram.

Table 5Exponential model parameters for Gilsocarbon density data.

Density exponential model parameters - Gilsocarbon

gðhÞ ¼ nþ c�1� e

ð�hLÞ�

Spine 3 Spine 6

L e Range 95.58 L e Range 277.66LP e Practical range 286.76 LP e Practical range 832.99c e Sill 42.02 c e Sill 66.90n e Nugget effect 1.73� 10�8 n e Nugget effect 12.37

Spine 9 Spine 12

L e Range 134.11 L e Range 76.76LP e Practical range 402.34 LP e Practical range 230.30c e Sill 45.31 c e Sill 75.04n e Nugget effect 1.67� 10�8 n e Nugget effect 1.56� 10�8


through the first two data points near the origin. Usually theplotted line will intersect the sill at 1/3 of the point where the lineintersects with the curve and is called range or correlation length.The equation for the exponential model is given by:

gðhÞ ¼ nþ C0h1� eð h

�LÞi

(6)

Fig. 14. Random fields for Gilsocarbon density data, the length of the sides of the cubes is 96field e cube e Spine 3, b) Random field e cube e Spine 6, c) Random field e cube e SpineRandom field e graphite brick e Spine 6, g) Random field e graphite brick e Spine 9, h) R

where n is the nugget effect, C0 is the sill, h is a distance, and L thecorrelation length or range.

3.3. Random field generator

Random fields can be created through several techniques bydifferent processes that can include spatial and/or time variables. Inthis study, we use a random field generator known as the Local

0mm and show the effect of different correlation lengths on density values. a) Random9, d) Random field e cube e Spine 12, e) Random field e graphite brick e Spine 3, f)andom field e graphite brick e Spine 12.

Table 6Exponential model parameters for Gilsocarbon Young's modulus in the x direction data.

Young's modulus exponential model parameters - Gilsocarbon


ð�hLÞ�

Young's modulus x direction

Spine 3 Spine 6

L e Range 133.80 L e Range 380.14LP e Practical range 401.40 LP e Practical range 1140.42c e Sill 0.03 c e Sill 0.12n e Nugget effect 0.007 n e Nugget effect 0.05

Spine 9 Spine 12


Young's modulus y direction

Spine 3 Spine 6


Spine 9 Spine 12


Young's modulus z direction

Spine 3 Spine 6


Spine 9 Spine 12



Average Subdivision (LAS) method to represent the spatial vari-ability of density and Young's modulus. Three dimensional randomfields are created in this paper, however, for the sake of simplicitythe one dimensional case is explained here. The random fieldgenerator for one-dimension uses a top-down recursive method. Tostart the procedure a general mean value is generated for theprocess. In the first stage, the region is subdivided into two equalsubdomains; the subdomains are assigned with a new value thatfulfils the condition that the values of each subdomain have to bethe mean of the global value (Parent). This process is repeated overand over until the desired refinement of the mesh is achieved(Fig. 11).

For the full description of the Local Average Subdivision Methodrandom field generator the reader is referred to Reference [30]. Thealgorithm to create 3D random fields for arbitrary geometries canbe found in Ref. [10].

The Local Average Subdivision Method requires four inputs: themean, standard deviation, correlation length and probability dis-tribution. These values can be calculated from raw data; the cor-relation length can be obtained from variography as was describedin the previous two subsections (Subsection 3.1 and 3.2). The datasets for Gilsocarbon and NBG-18 were tested for normality thatfollows a Gaussian distribution; both cases did not pass the

normality tests. In this research a log-normal distribution wasassumed for the random field generator. Inputs and outputs for thegeneration of realisations of a random field are summarised inFig. 12.

4. Results

4.1. Gilsocarbon density

The parameters and variograms obtained for the Gilsocarbondensity data are summarised in Table 4 and Fig. 13. Fig. 13 exhibitstypical behaviours of variograms in which the semivariance tendsto have lower values near the origin and increase with distanceuntil the values tend to reach a plateau (sill). Another indicator ofthe plateau of the variogram is the population variance included inthe plots. Fig. 13b and Table 5 show that the largest correlationlength or range is found in Spine 6; the lowest correlation length isfound in Spine 12.

The difference of correlation length and descriptive statistics fordensity can be better exemplified by the random fields illustrated inFig. 14. Two types of geometries are used to exemplify the vari-ability of the Gilsocarbon data, cubes of 960mm side and brickswith geometries similar to Hinkley Point B AGR channel brick

Fig. 15. Variograms for Gilsocarbon Young's modulus in x direction. a) Spine 3 variogram, b) Spine 6 variogram, c) Spine 9 variogram, d) Spine 12.


designs. Random fields for cubes through all the results sectionshave a side equal to 960mm. The geometry for the input parame-ters for the density random fields were extracted from Tables 1 and5. The practical range or practical correlation length was selected asthe parameter correlation length value for the random fields. Theuse of the practical correlation length as an input parameter for therandom field generator is used through all the posterior resultssections.

As was mentioned before, three main parameters control thematerial property values across a random field that are the mean,standard deviation and correlation length. The influence of theseparameters can be seen in the differences between random fieldsdepicted in Fig. 14. The mean controls the values around fluctua-tions in the random fields; this influence can be seen in all therandom fields of Fig. 14. The next term, the standard deviation canproduce larger fluctuations around the mean value if this value ishigh enough. An example of these variations can be found inFig. 14d and h. Finally, the correlation length can have two effectson the generation of random fields. Smaller correlation lengthsproduce a concentration of highly correlated sections of therandom field (Fig. 14c and d), whereas larger correlation lengthsproduce more smooth variations on the values of a random field

(Fig. 14b). Fig. 14eeh show how random fields can also be used tosimulate the spatial variability in complex geometries such as anAGR component. Much of the variation of density in the randomfield for the brick of Spine 12 data can be explained by the low valueof correlation length (Fig. 14h). A more homogenous distribution ofmaterial properties can be found in Fig. 14e and f.

4.2. Gilsocarbon Young's modulus

Table 6 provides the results for Young's modulus for the Gilso-carbon data set, the correlation length or range values for the samespine and different direction show similar values between them.The highest value for correlation length can be found in Spine 6;this trend is also found for the density value of Spine 6. In this casethe lowest correlation length value corresponds to Spine 9. Only thevariograms for Young's modulus in the x direction are included inthis study and are shown in Fig. 15.

As with the previous Gilsocarbon density results, a random fieldwas generated for each calculated correlation length (Fig. 16). Ingeneral, these random fields present similar patterns as the onesfound in Fig. 14. This resemblance is expected because of the rela-tionship between density and Young's modulus values [25,31]:

Fig. 16. Random fields for Young's modulus data in the x direction, the length of the sides of the cubes is 960mm and show the effect of different correlation lengths on Young'smodulus values. a) Random field e cube e Spine 3, b) Random field e cube e Spine 6, c) Random field e cube e Spine 9, d) Random field e cube e Spine 12, e) Random field e

graphite brick e Spine 3, f) Random field e graphite brick e Spine 6, g) Random field e graphite brick e Spine 9, h) Random field e graphite brick e Spine 12.

Table 7Exponential model parameters for NBG-18 density data.

Density exponential model parameters e NBG-18


ð�hLÞ�

Region 1 Region 2


Region 3 Region 4



graphite denser areas tend to be stiffer. The random fields forFig. 16a and b displays the smallest spatial variations of Young'smodulus, this behaviour is due to the small standard deviation inthe case of Fig. 16a and large correlation length value in the case ofFig. 16b. Random fields for AGR bricks are found in Fig. 16eeh. Themost homogeneous Young's modulus random field for AGR bricks isshown in Fig.16f; this is likely to be produce by the large correlationlength found in Spine 6. Closer inspection of the random fields for

graphite bricks indicate that the most spatial material variabilitycan be found in Fig. 16g. This behaviour is perhaps caused by thelow value for the correlation length in Spine 9.

4.3. NBG-18 density

NBG-18 density variography results are outlined in Table 7 andFig. 17. Compared to the Gilsocarbon density results, the NBG-18

Fig. 17. Variograms for NBG-18 density data. a) Region 1 variogram, b) Region 2 variogram, c) Region 3 variogram, d) Region 4 variogram.


ranges or correlation lengths are very similar between them as canbe seen in Table 7. Even though the variograms depicted in Fig. 17present different characteristics the range or correlation lengthare alike, because the plateaus of all the variograms tend to bearound 183mm. These results also suggest that the spatial vari-ability in NBG-18 is more consistent through the billet than theGilsocarbon data. Nevertheless, the low values of correlation length(183mm)mean that dissimilar density values can likely to be foundthrough the ends of the billet.

Further analysis was performed by producing a series of randomfields that represent the spatial material variability of the NBG-18density data (Fig. 18). All the random fields present a similarbehaviour on the distribution of material properties, but only therandom field for Region 1 have lower values due to the lower meanof this region. A very similar behaviour can be found in the case ofgraphite bricks; again, the random fields for regions 2 to 4 present ahomogenous spatial distribution around their means and Region 1shows a similar pattern but with lower density values.

4.4. NBG-18 Young's modulus

Parameters for the exponential model for the NBG-18 Young's

modulus data is summarised in Table 8. Notably, in Table 8 all therange or correlation length values are very similar and are also closeto the density correlation length. Again, this fact can be interpretedas a short-range-order distribution of material properties acrossthe NBG-18 billet. Variograms for NBG-18 Young's modulus data areshown in Fig. 19.

Fig. 19 shows the respective variograms of Young's modulus foreach region of the NBG-18 billet. A comparison between the pop-ulation variance plotted in Fig. 19 and range or correlation length ofTable 8 indicate that share similar values except for Region 4. Aswith previous results random fields were generated for the NBG-18Young's modulus data (Fig. 20). All the random fields represented inthis illustration show very similar spatial distributions of materialproperties and with a similar range of Young's modulus values.

5. Discussion

A methodology to extract and process material property data tocalculate the correlation length and other parameters necessary tocalibrate a random field for nuclear graphite components has beenpresented. The correlation length or range is an important factorthat measures the likelihood of finding similar material property

Fig. 18. Random fields for NBG-18 density data, the length of the sides of the cubes is 960mm and show the effect of different correlation lengths on density values. a) Random fielde cube e Region 1, b) Random field e cube e Region 2, c) Random field e cube e Region 3, d) Random field e cube e Region 4, e) Random field e graphite brick e Region 1, f)Random field e graphite brick e Region 2, g) Random field e graphite brick e Region 3, h) Random field e graphite brick e Region 4.

Table 8Exponential model parameters for NBG-18 Young's modulus data.

Young's modulus exponential model parameters e NBG-18


ð�hLÞ�

Region 1 Region 2


Region 3 Region 4



values within a certain distance. Descriptive statistics cannot fullydescribe the spatial changes of material properties through a me-dium even though the material has the same mean and standarddeviation. This can be proved by comparing the random fields of

Fig. 10, wherein these random fields share the same mean andstandard deviation but a very different spatial distribution of ma-terial property values.

One interesting finding of this research is that the spatial

Fig. 19. Variograms for NBG-18 Young's modulus. a) Region 1 variogram, b) Region 2 variogram, c) Region 3 variogram, d) Region 4 variogram.


distributions of material properties represented by the range orcorrelation length were different in the Gilsocarbon and NBG-18billet. Multiple correlation length values were found across thedifferent regions of the Gilsocarbon billet, showing that a morepronounced spatial variability can be expected in this billet. Onefactor that may contribute to the different values of correlationlength is extraction of samples around the edge of the billet. Den-sity and Young's modulus values tend to be higher and more con-stant through the edge of the billet. This edge effect produced largercorrelation lengths that might not be found at the interior of aGilscoarbon billet. In contrast, NBG-18 samples were taken acrossmultiple regions reducing the edge effect, producing consistentcorrelation lengths of about 180mm. This low correlation lengthvalues mean that spatial variations of mechanical properties can beexpected across a billet of NBG-18. An interesting consequencemight be the occurrence of “odd” or “outlier” data point in Arrhe-nius plots for graphite NBG-18 oxidation by air, as reported inde-pendently by Jo Jo Lee [32], Hans-Kemens Hinssen [33] and by S HChi [34]. Since the size of specimens in these studies was maximum25mm, it is very possible the ‘odd” specimens were cut from iso-lated local domains (islands) of low density graphite. Some exam-ples of this short-range correlation are the “islands” found in the

random field of Figs. 18 and 20.Spatial correlation is often overlooked in the normal assessment

of graphite billets and FEM studies of nuclear graphite. Spatialmaterial variability is a factor that may be contributor on thegeneration of defects and stress concentrations through the life-time of a graphite component. Furthermore, the understanding ofheterogeneity of graphite components could greatly benefit fromthe analysis of spatial variations proposed in this study. The cor-relation length calculations can also be a valuable tool to differen-tiate the effect of the grain size and proportions of filler and binderon the mechanical properties of graphite. Possibly very differentcorrelation lengths could be associated with the grain size andmixtures of binder and filler.

Recent FEM studies have considered the spatial distribution ofmaterial properties in their studies [10,35]. However, thesestudies do not address a technique or procedure to determinehow the spatial material variability would be determined. His-torical [12,13] and modern data [17,36,37] would allow one toutilise the geostatistics tools used in this paper to obtain a newinsight in the spatial variability of material mechanicalproperties.

Another advantage of including this methodology into the

Fig. 20. Random fields for Young's modulus data, the length of the sides of the cubes is 960mm and show the effect of different correlation lengths on Young's modulus values. a)Random field e cube e Region 1, b) Random field e cube e Region 2, c) Random field e cube e Region 3, d) Random field e cube e Region 4, e) Random field e graphite brick e

Region 1, f) Random field e graphite brick e Region 2, g) Random field e graphite brick e Region 3, h) Random field e graphite brick e Region 4.


assessment of graphite billets is that it could reduce the number ofspecific component types that need to be inspected. Similar cor-relation length values at different regions of different billets wouldmean a more even distribution of material properties acrossgraphite bricks and would ensure the quality of the material andmanufacturing process.

6. Summary and conclusions

This study demonstrates the use of random field theory in thecharacterisation of spatial material variability as a new approach formodelling the heterogeneity of graphite components. The corre-lation length is a parameter that can help to quantify the degree ofvariability of density, Young's modulus or any another materialproperty. Furthermore, FEM stress analysis can include randomfields to measure the influence of spatial material variability underthe reactor environment.

The main findings of this study research are described below:

1. Multiple correlation lengths values were found at the spines ofthe Gilsocarbon graphite billet. Practical correlation lengthsranged from 230.30 to 832.99mm for density, and204.63e1140.42mm for Young's modulus data.

2. Very similar correlation length values were obtained for theNBG-18 for both density and Young's modulus data. Moreover,similar material properties will be found around a 180mmdistance (the practical correlation length), meaning thatchanges of material properties can be expected at different lo-cations of this billet.

3. Random fields can be used to reproduce the mean, standarddeviation and correlation length in graphite components. Thesemodels can be included in FEM simulations to identify the in-fluence of spatial material variability. The generation of randomfields would also allow sensitivity studies to be performed onthe statistical parameters (mean, standard deviation, probabilitydistribution and correlation length) of different graphite grades.

4. The correlation length can be an important additional param-eter to determine the degree of heterogeneity in a nucleargraphite component.

Acknowledgements

The authors would like to acknowledge Gyanender Singh forproviding the geometry files of the prismatic graphite brick. OakRidge National Laboratory is managed by UT-Battelle, LLC underContract No. DE-AC05-00OR22725 for the U.S. Department of


Energy. This work was supported by the ORNL Postdoctoral Per-formance Development program.

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