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Effective thermal conductivity of heterogeneous materials: calculation methods and application to different microstructures
Dragos M Staicu, Dominique Jeulin, Michel Beauvy, Michel Laurent, Christine Berlanga, Norbert Negrello, Dominique Gosset

First of all, three different methods are presented for calculating the macroscopic thermal conductivity of composite materials: numerical methods, predictive analytical methods, and the bounds obtained by variational methods. Their predictions are then compared with measured results for several types of nuclear material having different microstructures and constituent conductivity ratios. The numerical calculations, which use the actual microstructure, are in principle the most accurate, but can generally be performed in two dimensions only, using the image of a section. The two-dimensional (2D) numerical results provide a good evaluation of the conductivity when the matrix and inclusions have similar conductivities. If this is not the case, they underestimate the true three-dimensional (3D) value; a 2D/3D conversion method must therefore be determined by the use of one of the models from one of the other categories. The predictive analytical results, obtained with idealised microstructures, are accurate provided the microstructure is simple or if the conductivities of the constituents are similar. The upper bounds obtained by variational methods correspond to microstructures close to the actual microstructure, and are the most accurate when the difference in conductivity between the constituents is large and the volume fraction of the inclusions is high. The choice of the most suitable model can be made by comparing the 2D model predictions with the results of the 2D numerical calculation results.

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