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 eγ (e > 0 and β > 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 e2 (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 eβ with 0 < e ⩽ 1.

中文翻译：

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双孔型三相复合材料的均质化

在这项工作中，我们考虑具有三相的周期性复合材料中的扩散问题：基体、纤维和中间相。介质的热导率在纤维的横向上以大小为 eγ（e > 0 和 β > 0）的周期周期性变化。此外，我们假设界面材料的电导率和纤维中材料的各向异性对比度为 e2 级（所谓的双孔型缩放），而基体材料的电导率为 1 级。通过为各向异性域引入部分展开算子，我们确定了极限问题。特别是，我们证明了只有当微观结构长度尺度为 eβ 且 0 < e ⩽ 1 时，才能捕捉到界面特性对均质模型的影响。