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homogenization; three-phase composite; unfolding operator; double-porosity type
In this work we consider a diffusion problem in a periodic composite having three phases: matrix, fibers and interphase. The heat conductivities of the medium vary periodically with a period of size $\varepsilon ^\beta $ ($\varepsilon >0$ and $\beta >0$) in the transverse directions of the fibers. In addition, we assume that the conductivity of the interphase material and the anisotropy contrast of the material in the fibers are of the same order $\varepsilon ^2$ (the so-called double-porosity type scaling) while the matrix material has a conductivity of order $1$. By introducing a partial unfolding operator for anisotropic domains we identify the limit problem. In particular, we prove that the effect of the interphase properties on the homogenized models is captured only when the microstructural length scale is of order $\varepsilon ^\beta $ with $0<\beta \leq 1$.
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