Micromechanics is a useful tool for the modelling and analysis of the behaviour of heterogeneous materials. One of its main objectives is to estimate their macroscopic effective properties from their microstructural features (phase properties, inclusions distribution and geometry, ...). This topic is even more interesting when there is a lack of experimental data. Many studies dedicated to micromechanics mostly considers only the elastic behaviour. But thermal, electrical and piezoelectric properties are equally important and are not so much investigated. Based on the works of Eshelby on the single-inhomogeneity problem, several homogenization schemes have been developed to derive the effective behaviour of composite materials. Through the Eshelby tensor, it is indeed possible to account for inclusions with different properties and geometries. The question of interactions between phases can be also addressed with such modelling approach.

In the case of microcracked materials, various studies modelled the crack as a flat oblate ellipsoid with aspect ratio tending to zero. The macroscopic behaviour changes based on the state of the crack (open or closed) makes the task even more challenging. This unilateral effect is taken into account by defining different properties for the crack according to its status. For example, in elasticity problems, the open cracks are considered as traction free defects with zero elasticity, whereas closed cracks can be represented by some fictitious material with specific properties.

We intend to derive the closed-form expressions of the effective thermal conductivity and resistivity tensors of microcracked media at fixed damage state. This is possible thanks to the mathematical analogy between elasticity and steady state heat conduction problems. We consider an initially isotropic matrix weakened by flat, randomly distributed microcracks. Microcracks are described by their orientation (given by a unit vector), scalar density and conductivity (resp. resistivity). Due to the orientation of defects, the resulting material is macroscopically anisotropic. To determine the effective thermal conductivity (resp. resistivity) we use the temperature gradient (resp. heat flux) based formulations i.e. we impose a uniform macroscopic temperature gradient (resp. heat flux) at the outer boundary of the RVE. The estimates are calculated in both dilute and Mori-Tanaka schemes. The study is done for a single family of parallel defects that can be either open or closed friction-less, the unilateral effect being taken into account through the definition of the cracks conductivity (resp. resistivity).

Influences of assumptions regarding boundary conditions and interactions between cracks are discussed. Moreover, results highlight the contribution of the unilateral effect of microcracks to thermal conduction properties. This will help in developing a brittle damage model accounting for thermomechanical coupling, especially the influence of damage on steady state heat transfer.