Effective thermal conductivity of a composite with thermo-sensitive constituents and related problems

The paper focuses on the problems related to homogenization procedure for thermal conductivity of a composite with thermo-sensitive constituents (i.e. materials with conductivities dependent on temperature). It is shown that in the simplest case of non-linearity, when thermo-sensitive inhomogeneity is embedded into linearly-conductive matrix, Eshelby theorem (Eshelby, 1957, 1961) does not hold – remotely applied uniform heat flux yields a non-uniform one inside the inhomogeneity. However, in the case when both – the matrix and the inhomogeneity – are thermo-sensitive and their conductivities are proportional to each other, Eshelby theorem for heat flux does hold. For materials of this type, the concept of resistivity contribution tensors is formulated that allows one to generalize the main homogenization schemes used in micromechanics for this special case of non-linearity. The requirement of proportionality holds for many important material systems, including, in particular, porous thermo-sensitive materials or ones reinforced with superconductive inhomogeneities.