Application of a micromechanics model to the overall properties of heterogeneous graphite C. Berre a, * , P.M. Mummery b , B.J. Marsden a , T. Mori b , P.J. Withers b a School of Mechanical, Aerospace and Civil Engineering, University of Manchester, P.O.Box 88,Sackville Street, Manchester M60 1QD, UK b School of Materials, University of Manchester, Grosvenor Street, Manchester M1 7HS, UK a r t i c l e i n f o PACS: 02.60.Cb 62.20.Dc 81.05.Uw a b s t r a c t This paper deals with the overall properties of polycrystalline graphite, a material mainly composed of voids and dense inhomogeneities embedded in a less dense matrix. First, we examine the overall average elastic properties and conductivities of such a material. Second, we evaluate the void shape effects on the overall Young’s modulus. Finally, we compare the results obtained from the analytical model with exper- imental data from radiolytic oxidation of graphite. Ó 2008 Published by Elsevier B.V. 1. Introduction Because ofits very good thermo–mechanicalproperties in a large range of temperatures, graphite is used for many industrial applications and particularly in the nuclear industry. Its manufac- ture involves complex methods of mixing and baking at high temperatures, resulting in a heterogeneous material made of coke filler particles,a coal–tar pitch binder matrix and pores of various sizes [1,2].In the UK, advanced gas-cooled reactors (AGRs) use a dense and near isotropic type of graphite as a moderator and as a major structural component. During service, the microstructure of graphite is subjected to neutronic irradiation and radiolytic oxida- tion, leading to important microstructural changes. It is now well established that these changes are related to the bulk mechanical properties of the material [2,3]. These property changes and their relationships to the microstructure have been since long the sub- ject of many studies and considerable effort has been made on the development of porosity due to radiolytic oxidation [4,5]. The aim of this paper is to use an analytical approach to under- stand and to evaluate the microstructural changes due to increas- ing pore volume fractions and their relationships with the overall mechanical properties of heterogeneous graphite. Eshelby’s paper [6] is a basic general theory of micromechan- ics of an inhomogeneity problem. Several extended theories have been proposed since Eshelby [7]. Among them, a mean field method has been used widely for its simplicity. In the context of the present study, a paper by Taya and Chou [8] is most signif- icant,since the paper examined the overall elastic constants of a composite having two types of inhomogeneities.However, as shown later,the basic equations to derive the overall elastic con- stants in their study lack one critical point. Thus,this paper first presents our understanding of micromechanics of a material hav- ing two types of inhomogeneities.In the main development of analysis,we will assume that the shape ofan inhomogeneity is ellipsoidal.Firstly, a spherical shape is assumed for two types of the inhomogeneities in the actual calculations, since the inhomo- geneities we encounterin the type of graphite considered are nearly spherical or particulate.In addition, we will adopt the method using the extra strain due to the inhomogeneities in or- der to calculate the overallelastic constants.This strain method is different from that originally given by Eshelby, but can be more easily accommodated into the final expressions of the overall constants.However, we will show that the two methods are equivalent. Next, the overall thermo–mechanicalproperties of the same material are evaluated using the same principle and the influence of the stiffness of the particles and the shape of the pores on the elastic properties is shown. Finally, the numeri- cal calculations of the overall Young’s modulus for increasing pore volume fractions are compared with radiolytic experimental data from the literature. 2. Analysis 2.1.Elastic body consisting of three isotropic phases Eshelby demonstrated that the disturbance of the stress field, created by the presence of one inhomogeneity in an infinitely ex- tended and elastic matrix,can be reproduced by an equivalent inclusion.The equivalent inclusion must possess the same shape as the inhomogeneity it represents, and the same properties as the surrounding matrix in which it is embedded.The local ‘stress-free’strain within the inclusion is here referred to eigen- 0022-3115/$ - see front matter Ó 2008 Published by Elsevier B.V. doi:10.1016/j.jnucmat.2008.07.015 * Corresponding author.Tel.: +44 (0) 161 275 4437. E-mail address: Christophe.Berre@postgrad.manchester.ac.uk (C. Berre). Journal of Nuclear Materials 381 (2008) 124–128 Contents lists available at ScienceDirect Journal of Nuclear Materials j o u r n a l homepage: w w w . e l s e v i e r . c o m / l o c a t e / j n u c m a t