In this paper, we use approximate solutions of NematNasser et al. to estimate the effective conductivity of two-phase periodic composites with non-overlapping spherical inclusions. Systems with different inclusion distributions are considered: cubic lattice distributions (simple cubic, body-centred cubic and face-centred cubic) and random distributions. The effective conductivities of the former are obtained in closed form and compared with exact solutions from the fast Fourier transform-based methods. For systems containing randomly distributed spherical inclusions, the solutions are shown to be directly related to the static structure factor, and we obtain its analytical expression in the infinite-volume limit.
International audienceThis study deals with the heat conduction within a medium containing cracks that are assumed to be perfect insulators. Multi-region boundary element approach is employed to obtain a boundary singular integral equation governing the steady state thermal transfer within this medium. This equation presents the temperature field within the whole cracked body as a function of temperature and rate of heat flow on the domain's boundary and temperature discontinuity across the cracks. For the particular case of an infinite domain under far-field condition, the temperature field solution is only a function of the cracks tem-perature's discontinuity. The basic problem of a single crack in an infinite domain is investigated and a closed-form solution is derived for a crack of elliptic plane from this analysis. This solution is the key issue to estimate the effective thermal conductivity of the whole domain by coupling with the classical homogenization schemes. The arbitrary crack form is covered up by using the excluded volume definition. Estimations of effective thermal conductivities stemming from diluted, differential and self-consistent approaches are compared to numerical solution obtained by the finite volume modeling that is available in literature. This comparison shows that the self-consistent scheme is the most appropriate model to estimate the thermal conductivity of materials containing cracks
This paper aims at predicting effective transport properties of fractured porous media based on the information of cracks distribution within the material. The porous materials are assumed to contain aligned insulating and superconductive cracks, arranged in parallel layers. Two types of cracks distributions are considered: periodic and random distributions. In the former periodic case, the estimates are analytically derived from the approximation of polarization integral equations, and compare well with a numerical solution. In the latter random case, the estimates show explicit connections to planar structure factors of hard disks, a statistical quantity of phases distribution in Fourier space. Different random ensembles of hard disks are also examined to study how they affect the effective permeability of the material.
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