Homogenization of a Three-Phase Composites of Double-Porosity Type
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Czechoslovak Mathematical Journal
29 pp
Online first
HOMOGENIZATION OF A THREE-PHASE COMPOSITES OF DOUBLE-POROSITY TYPE Ahmed Boughammoura, Yousra Braham, Monastir Received April 1, 2019. Published online July 1, 2020.
Cordially dedicated to Professor Mongi Mabrouk Abstract. 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 εβ (ε > 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 ε2 (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 εβ with 0 < β 6 1. Keywords: homogenization; three-phase composite; unfolding operator; double-porosity type MSC 2020 : 35B27, 35B45, 35K55, 35K65, 76S05
1. Introduction It is well known that the effective mechanical and thermal properties of a combination of fibers, which have highly-anisotropic characteristics, with matrix materials of a completely different nature is strongly influenced by the presence of an inevitable interphase between these two phases (see e.g. [8] and [12]). As a consequence, the fiber-matrix interphase has a crucial impact on the behavior and properties of the fiber-reinforced composites. In recent years, by applying numerical homogenization techniques, several studies have shown the influence of the interphase material parameters on the overall mechanical behavior of fibrous composites (see e.g. [2], [10], [11]). However, it should DOI: 10.21136/CMJ.2020.0151-19
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be noted that a few mathematical results have been established rigorously in the analysis of those effects. To the author’s knowledge, probably one of the first papers performing a rigorous mathematical analysis of the effects of the interphase in the overall behavior of fiberreinforced composites was published in 2009 by the first author (see [3]). Using the two-scale convergence method, the author studied the homogenization of a heat transfer problem in a fiber-reinforced composite taking into account the combined effects of the fiber-matrix interphase together with the high anisotropy of the material in the fibers. More precisely, he considered a degenerate elliptic-parabolic equation in a composite, whose physical properties oscillate rapidly on the scale of order ε (ε is a small dimensionless parameter). The most interesting results concerned the critical case, where the conductivity of the interphase material and the anisotropy contrast in the fiber are of the same order ε2 (the so-called double-porosity type scaling). Those results showed, in particular, that the homogenized problem is an integr
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