Heat conduction and thermal conductivity of 3D cracked media

This 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 temperature'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.