Elsevier

Journal of Nuclear Materials

Volume 511, 1 December 2018, Pages 91-108
Journal of Nuclear Materials

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

https://doi.org/10.1016/j.jnucmat.2018.09.008Get rights and content

Abstract

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 different material property values between different components. Variations in material properties and irradiation effects can produce stress concentrations and diverse mechanical responses in a nuclear reactor graphite core. In order to characterise the material variability, geostatistical techniques called variography and random field theory were adapted for studying the density and Young's modulus of a billet of Gilsocarbon and NBG-18 graphite grades. Variography is a technique for estimating the distance over which material property values have significant spatial correlation, known as the scale of fluctuation or spatial correlation length. The paper uses random field theory to create models that mimic the original spatial and statistical distributions of the original data set. This study found different values of correlation length for 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 to reproduce the spatial patterns and values found in the original data.

Introduction

Graphite components are deployed in multiple present-day nuclear power stations including the Advanced Gas-cooled Reactors (AGR) in the UK and will be part of Generation IV designs such as the Very High Temperature Reactor (VHTR) and some Molten Salt Reactors (MSR) concepts. The graphite core of these reactors designs serve as a moderator of fast neutrons, repository for fuel, and to provide structural support [1]. The reactor environment promotes several ageing mechanisms in the graphite components that lead to material property and dimensional changes. Examples of these mechanisms are the irradiation dimensional changes, irradiation creep, thermal strains and oxidation. The combination of these ageing and degradation mechanisms may lead to the distortion and internal stresses of individual components that in time can result in the formation and propagation of cracks. When enough graphite components crack or deform, the geometry of the core may interrupt the normal operations of a nuclear reactor, such as refuelling operations and the cooling thermodynamics of the graphite core. Thousands of graphitic bricks form the reactor core, the mechanical response of each of these components is highly dependent on the fluence, temperature profile, coolant and mechanical properties of the specific graphite grade. A certain degree of mechanical property variability is expected to be found inside and in-between billets. This variability is normally not accounted by the assessment and inspections on the graphite core.

Part of the routine assessments and predictions of the lifetime of graphite components include Finite Element Method (FEM) simulations. Assessments based on FEM models require an extensive database of the unirradiated and irradiated response of graphite. Some examples of FEM analysis for AGR components can be found in literature [[2], [3], [4]]. In general, these models include the effects of thermal expansion, irradiation-induced dimensional changes, irradiation creep and other ageing mechanisms that may alter the constitution of a graphite component. The structural integrity studies of graphite components are usually supported by stress analysis and other calculations that estimate the failure rate and lifetime of the components. In addition to stress analysis, several fracture mechanics techniques have been implemented to analyse the crack evolution in nuclear graphite components [[5], [6], [7], [8], [9]]. Traditionally FEM and continuum damage techniques use the mean values of material properties to perform their analyses. Unfortunately, by only using the mean value of the material properties all the spatial and statistical information such as standard deviation and probability distribution are lost. A previous study that includes spatial material variability into FEM has demonstrated that heterogeneity of material properties in AGR reactors can lead to stress concentrations [10].

Quantifying the degree of heterogeneity of graphite components is essential to predict possible differences on the failure rate of graphite components. Inconsistencies in material properties may lead to the generation of stress concentrations – this effect can be produced by the combination of a temperature gradient and a significant heterogeneity of material properties in the graphite bricks [11]. A similar effect can be expected to be produced by the combination of irradiation-induced dimensional change and material property variation within a single graphite component.

The objectives of this research are to characterise the material's spatial variability through geostatistical techniques and reproduce the spatial fluctuations of two grades of nuclear graphite through mathematical models called random fields. The characterisation of spatial fluctuations only requires additional calculations that provide a new insight on the variations of materials properties within a billet. Alternatively, random field realisations reproduce models of the spatial fluctuations found in graphite allowing the modeller and designers to simulate different scenarios. This research focuses on the density and dynamic Young's modulus parameters of Gilsocarbon and NBG-18, although other material properties of interest can also be studied using this methodology.

Several studies have shown the presence of heterogeneity of material properties in graphite components. These studies have been carried out for Gilsocarbon, the moderator of AGRs [[12], [13], [14]], H-451 used at Peach Bottom and Ft. St. Vrain reactors (USA) [15,16] and NBG-18 [17] a modern graphite grade proposed for future VHTRs.

Gilsocarbon is a medium grain graphite composed of spherical onion-shaped filler particles that are moulded during the manufacturing process [18]. The combination of moulding and spherical particles in Gilsocarbon result in an isotropic or semi-isotropic mechanical behaviour. The name of Gilsocarbon grade was given to the family of graphite grades designed for the AGRs in the UK. All these grades differ from each other as they were manufactured by two different companies, Anglo Great Lakes Corporation Limited (AGL) and Union Carbide. Although all these grades where manufactured in similar conditions, all have slightly different grain sizes and microstructure. Several studies were conducted on the possible variations of properties among these graphite grades and within single billets. The first published measurements on the spatial variability of the physical and mechanical properties in Gilsocarbon within a single billet were conducted by Preston [13,14]. Approximately half of a billet was sectioned to measure several types of physical and mechanical properties. These measurements include density, electrical resistivity, coefficient of thermal expansion, four-point bending strength, compressive strength, Young's modulus, tensile strength, thermal conductivity, open pore volume, closed pore volume and Poisson's ratio. These studies confirmed the spatial variability of material's physical and mechanical properties. However, the data by Preston are not ideal for calibrating a random field for the purposes of this study. Therefore, another source of density and dynamic Young's modulus was used for the calculations of this study [19].

The second material examined here, NBG-18, was a candidate grade 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 vibrationally moulded with a medium grain size (about 1.6 mm) [20,21]. An extensive qualification research program was developed for this grade at Idaho National Laboratory. Part of this program focused on the characterisation of a single unirradiated billet of NBG-18 and is also a guideline for future studies for other grades of graphite. For this research a billet of this grade was sectioned into 770 specimens that were used to measure the density, compressive strength, tensile strength, Young's modulus, shear modulus and other mechanical properties. The mechanical performance of this graphite grade under irradiation is also currently being investigated under the Advanced Graphite Creep programs (AGC) AGC-1 [21] and AGC-2 [22].

Section snippets

Gilsocarbon data

The density and Young's modulus variability of material properties were obtained by sectioning a billet of Gilsocarbon provided by EDF Energy Nuclear Generation Ltd. This billet was manufactured by Union Carbide for the Torness/Heysham II reactors (heat 4904; serial number 07V3632). The samples were sectioned from a billet with dimensions of 930 mm in height and 470 mm in diameter (Fig. 1a). The geometry and dimensions of an AGR reactor can vary depending on the nuclear power station. A typical

Methods

In order to improve the understanding of spatial material variability within a data set it is necessary to create and interpret a variogram. A variogram estimates the shape, correlation length or range, and direction of spatial autocorrelation. The process to create a variogram can be divided into two steps: (1) the first is to calculate an experimental variogram and (2) the second step is to derive a variogram estimator from the experimental variogram. These two steps are known collectively as

Gilsocarbon density

The parameters and variograms obtained for the Gilsocarbon density data are summarised in Table 4 and Fig. 13. Fig. 13 exhibits typical behaviours of variograms in which the semivariance tends to have lower values near the origin and increase with distance until the values tend to reach a plateau (sill). Another indicator of the plateau of the variogram is the population variance included in the plots. Fig. 13b and Table 5 show that the largest correlation length or range is found in Spine 6;

Discussion

A methodology to extract and process material property data to calculate the correlation length and other parameters necessary to calibrate a random field for nuclear graphite components has been presented. The correlation length or range is an important factor that measures the likelihood of finding similar material property values within a certain distance. Descriptive statistics cannot fully describe the spatial changes of material properties through a medium even though the material has the

Summary and conclusions

This study demonstrates the use of random field theory in the characterisation of spatial material variability as a new approach for modelling the heterogeneity of graphite components. The correlation length is a parameter that can help to quantify the degree of variability of density, Young's modulus or any another material property. Furthermore, FEM stress analysis can include random fields to measure the influence of spatial material variability under the reactor environment.

The main

Acknowledgements

The authors would like to acknowledge Gyanender Singh for providing the geometry files of the prismatic graphite brick. Oak Ridge National Laboratory is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 for the U.S. Department of Energy. This work was supported by the ORNL Postdoctoral Performance Development program.

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    This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

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