# Institute of Mathematics

## Homogenization of a three-phase composites of double-porosity type

#### Ahmed Boughammoura, Yousra Braham

###### Received April 1, 2019.   Published online July 1, 2020.

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 $\varepsilon^\beta$ ($\varepsilon>0$ 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 $0<\beta\leq1$.
Keywords:  homogenization; three-phase composite; unfolding operator; double-porosity type
Classification MSC:  35B27, 35B45, 35K55, 35K65, 76S05
DOI:  10.21136/CMJ.2020.0151-19

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References:
[1] G. Allaire, F. Murat: Homogenization of the Neumann problem with nonisolated holes. Asymptotic Anal. 7 (1993), 81-95. DOI 10.3233/ASY-1993-7201 | MR 1225439 | Zbl 0823.35014
[2] H. Berger, S. Kurukuri, S. Kari, U. Gabbert, R. Rodriguez-Ramos, J. Bravo-Castillero, R. Guinovart-Diaz: Numerical and analytical approaches for calculating the effective thermo-mechanical properties of three-phase composites. J. Thermal Stresses 30 (2007), 801-817. DOI 10.1080/01495730701415665
[3] A. Boughammoura: Homogenization of a degenerate parabolic problem in a highly heterogeneous medium with highly anisotropic fibers. Math. Comput. Modelling 49 (2009), 66-79. DOI 10.1016/j.mcm.2008.07.034 | MR 2480033 | Zbl 1165.80302
[4] A. Boughammoura: Homogenization of an elastic medium having three phases. Ric. Mat. 64 (2015), 65-85. DOI 10.1007/s11587-014-0211-y | MR 3355900 | Zbl 1320.35036
[5] D. Cioranescu, A. Damlamian, G. Griso: Periodic unfolding and homogenization. C. R., Math., Acad. Sci. Paris 335 (2002), 99-104. DOI 10.1016/S1631-073X(02)02429-9 | MR 1921004 | Zbl 1001.49016
[6] D. Cioranescu, A. Damlamian, G. Griso: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40 (2008), 1585-1620. DOI 10.1137/080713148 | MR 2466168 | Zbl 1167.49013
[7] D. Cioranescu, P. Donato: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and Its Applications 17. Oxford University Press, Oxford (1999). MR 1765047 | Zbl 0939.35001
[8] L. T. Drzal: Interfaces and interphases. ASM Handbook, Volume 21. Composites D. B. Miracle, S. L. Donaldson ASM International, Ohio (2001), 169-179.
[9] J. Franců, N. E. M. Svansted: Some remarks on two-scale convergence and periodic unfolding. Appl. Math., Praha 57 (2012), 359-375. DOI 10.1007/s10492-012-0021-z | MR 2984608 | Zbl 1265.35017
[10] S. Kari, H. Berger, U. Gabbert, R. Guinovart-Diaz, J. Bravo-Castillero, R. Rodrigues-Ramos: Evaluation of influence of interphase material parameters on effective material properties of three phase composites. Composites Sci. Techn. 68 (2008), 684-691. DOI 10.1016/j.compscitech.2007.09.009
[11] Y. Mikata, M. Taya: Stress field in and around a coated short fiber in an infinite matrix subjected to uniaxial and biaxial loadings. J. Appl. Mech. 52 (1985), 19-24. DOI 10.1115/1.3168996
[12] B. Pukánszky: Interfaces and interphases in multicomponent materials: Past, present, future. European Polymer J. 41 (2005), 645-662. DOI 10.1016/j.eurpolymj.2004.10.035

Affiliations:   Ahmed Boughammoura, University of Monastir, Higher Institute of Computer Sciences and Mathematics of Monastir, Corniche Avenue, BP 223, Monastir 5000, Tunisia, e-mail: ahmed.boughammoura@gmail.com; Yousra Braham, University of Monastir, Faculty of Sciences of Monastir, Avenue of the environment, 5019 Monastir, Tunisia, e-mail: brahamyosraa@gmail.com

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